Classification and Orbit Equivalence Relations

Greg Hjorth

Author address: Department of Mathematics, University of California at Los An-

geles, Los Angeles, California, 90095-1555

E-mail address: greg@math.ucla.edu

1991 Mathematics Subject Classification. Primary 04A15; Secondary 22A05, 54H05, 54H20.

To Noela and Bob Hjorth

Contents

Preface xi

Chapter 1. An outline 1 1.1. Some specific classification problems 1 1.2. The form of this book (and one time paper) 2 1.3. Acknowledgments 3

Chapter 2. Definitions and technicalities 5 2.1. Polish groups and Polish spaces 5 2.2. Equivalence relations 15 2.3. Spaces of countable structures 18 2.4. Baire category methods 23

Chapter 3. Turbulence 37 3.1. Generic ergodicity 37 3.2. The definition of turbulence 41 3.3. Examples 52 3.4. Historical remarks 55

Chapter 4. Classifying homeomorphisms 59 4.1. Definitions and remarks 59 4.2. Classification in dimension 1 61 4.3. Non-classification in dimension 2 66 4.4. Remarks and connections 74

Chapter 5. Infinite dimensional group representations 75

Chapter 6. A generalized Scott analysis 81 6.1. A preliminary discussion of a specific case 81 6.2. The general case 93 6.3. A counterexample 108 6.4. A different direction 111

Chapter 7. GE groups 115 7.1. More on Polish groups; Glimm-Effros; G_{δ} orbits 115 7.2. Invariantly metrizable and nilpotent are GE 128 7.3. Dynamic changes in topologies 134 7.4. Products of locally compact groups 139 7.5. CLI groups have the weak Glimm-Effros property 145

Chapter 8. The dark side 149

x CONTENTS

Chapter 9. Beyond Borel 155 9.1. Two theorems by transfinite changes in topologies 155 9.2. Cardinality in L(ℝ) 169

Chapter 10. Looking ahead 177 Appendix A. Ordinals 181 Appendix B. Notation 185 Bibliography 189 Index 193

Preface

What does it mean to classify the equivalence classes of some equivalence relation?

We have some tangible space whose points are very definite. Let X be the space. Maybe it is the reals. Or the complex numbers. Or perhaps all groups with underlying set ℕ. In any case, a concrete object.

In addition there is an equivalence relation E and we consider the quotient X/E. When should we judge this quotient object consisting of all equivalence classes to be comprehensible? When should we allow that the set of all equivalence classes, {[x]_{E} : x ∈ X}, is classifiable?

There is no absolute agreement on what constitutes a good classification theorem. It is necessarily a vague concept. But even granting its vagueness, there are probably some attempts one could dismiss out of hand.

For instance, assigning the equivalence class of x as a complete invariant for each point in the space is hardly satisfactory, since it provides us with no progress at all. The complete invariants should be objects we feel to be reasonably well understood – or at least, less mysterious than the equivalence classes with which we began.

Similarly we do have some standards concerning how the invariants can be produced. Simply appealing to the axiom of choice to well order the equivalence classes of E, and then using some corresponding well order of ℝ to assign real numbers as complete invariants does not constitute a satisfactory system of classification, even though the complete invariants in this case are indeed well understood. The system of classification should assign invariants to the points in the space X based on their intrinsic properties; even though the properties we may use could be highly subtle or extremely complex, we would have much greater respect for a system of classification that is fantastically difficult than one that just pulls down the axiom of choice and then goes home to bed.

In general terms a complete classification of E should consist of a reasonably intrinsic or definable function

θ : X → I

from X to a reasonably well understood collection of invariants I so that for all x and y in X

xEy ⇔ θ(x) = θ(y).

That much is a platitude. Beyond this there is great disagreement.

Example 0.1. Low dimensional topology For compact orientable surfaces a complete invariant can be obtained by simply counting the number of handles, and moreover, this invariant can be produced in a recursive or computable fashion from a finite triangularization of the manifold; for non-orientable surfaces the structure theorem is more complicated, but again the invariant can be taken to be a finite object – indeed it can be represented by a natural number – and may be computed from a triangularization using a purely finite search. ([12], [46].) In higher dimensions it is known from [64] that the isomorphism relation is not computable in the sense of Turing machines, and this is sometimes taken by topologists as indicating that there is no satisfactory system of complete invariants for higher dimension manifolds.

This is an extremely restrictive definition. Here a classification is a computable function

θ : X → ℕ

where X is a collection of finite triangularizations of manifolds so that x, y ∈ X code homeomorphic manifolds if and only if θ(x) = θ(y). This is the most stringent notion of classification: The invariants are finite, the process by which we assign them finitary.

Example 0.2. Linear algebra A rather different example is suggested by linear algebra. We may reasonably regard two unitary operators over a finite dimensional (complex) Hilbert space as somehow equivalent if there is a third that conjugates them – so in this sense we obtain an orbit equivalence relation on the space of all unitary operators on (ℂ^{n}, ·). A complete invariant for an operator is given by its finite set of eigenvalues, considered up to multiplicity, and it is always possible to encode finitely many complex numbers by a single real; thus we can assign in a Borel fashion to each element of U_{n}, the group of unitary operators over (ℂ^{n}, ·), a real number as a complete invariant.

Equivalence relations which in a Borel fashion allow real numbers as complete invariants are known as smooth or tame; thus E_{G} on X is smooth if there is a Borel

θ : X → ℝ

that reduces E_{G} to equality on ℝ, in the sense

xE_{G}y ⇔ θ(x) = θ(y).

[30], like [14] that it followed, takes classifiable to mean smooth.

Example 0.3. Ergodic theory Consider the classification problem for measure preserving transformations of the unit interval. It is natural to say that π_{1}, π_{2} : [0, 1] → [0, 1] are equivalent or isomorphic if there is some measure preserving bijection σ : [0, 1] → [0, 1] with

σ ∘ π_{1} ∘ σ^{−1} = π_{2} a.e.

This equivalence relation arises from a group action – the action of this group on itself by conjugation. In two special cases there are classification theorems.

Ornstein’s classification of Bernoulli shifts in [71] provides a proof that isomorphism on this class of measure preserving transformations is smooth. One can assign to each Bernoulli shift its entropy – a real number that completely classifies a Bernoulli shift up to isomorphism. This is one of the most celebrated theorems of ergodic theory, but it is not true that the only notion of classification in ergodic theory is that of being smooth or reducible to the equality relation on ℝ. A rather more generous notion of classification is suggested by a classic paper of Halmos and von Neumann.

In [29] Halmos and von Neumann show that for discrete spectrum measure preserving transformations we can assign a countable collection {c_{i}(π) : i ∈ ℕ} of complex numbers that completely describe the equivalence class of π. This assignment is indeed natural, since it arises from taking the eigenvalues of the form λ ∈ ℂ for which there is some non-zero f ∈ L^{2}([0, 1]) with f ∘ π = λf a.e.

The notion of classification here cannot be reduced to that of 0.2. For instance [20] shows that there is no reasonable method for representing countably infinite sets of real or complex numbers by single points in ℝ; indeed without appeal to the axiom of choice we may find it impossible to produce any injection from P_{ℵ_{0}}(ℝ) (the set of all countable collections of reals) to ℝ. The conjugacy relation on discrete spectrum measure preserving transformations is non-smooth, and yet the perspective of say [82] would uphold this as a complete classification for discrete spectrum measure preserving transformations.

Example 0.4. Locally compact group actions In [49] Kechris proves that the orbit equivalence relations induced by locally compact Polish groups are all reducible to countable equivalence relations. If we have locally compact G acting continuously on Polish X, with orbit equivalence relation E_{G}, then we can find a Borel equivalence relation F all of whose equivalence classes are countable such that we may assign to each point x ∈ X some corresponding θ(x) so that

x_{1}E_{G}x_{2} ⇔ θ(x_{1})F θ(x_{2}).

This result can be viewed as a classification theorem for orbit equivalence relations induced by locally compact group actions. We may assign to each x ∈ X the countable set {y : yF θ(x)} to obtain an invariant similar in structure to the Halmos-von Neumann spectral invariants.

Example 0.5. Point set topology The Cantor-Bendixson derivation as described in [52] can be used to provide a classification for countable compact metric spaces. At the first stage we remove the isolated points. At the next we remove the isolated points from the remaining space, and so on, through however many countable ordinals as are needed. This analysis provides a complete invariant of the space consisting of two parts: The ordinal length of this process, along with the number of points left standing before the termination at the final stage. Two countable compact metric spaces will be homeomorphic if and only if their Cantor-Bendixson derivations require the same ordinal number of steps and at the penultimate moment they share the same finite number of points remaining.

Example 0.6. Abelian group theory In [21] ordinals also enter stage in the famed Ulm invariants from abelian group theory. In essence the Ulm invariants are bounded subsets of ℵ_{1}, the first uncountable ordinal, that completely describe the isomorphism type of a countable torsion abelian group.

Example 0.7. Topological dynamics The authors of [24] classify so called minimal Cantor systems up to strong orbit equivalence by assigning countable ordered abelian groups. Two continuous

φ_{1} : X_{1} → X_{1},
φ_{2} : X_{2} → X_{2}

which are minimal in the sense of having no non-trivial closed invariant sets and are Cantor in the sense of X_{1}, X_{2} being compact, uncountable, and zero-dimensional metric spaces, are said to be strong orbit equivalent if there is a homeomorphism F : X_{1} → X_{2} which respects the orbits structure set wise and with the resulting conjugation suffering at most a single discontinuity – so that if m : X_{1} → ℤ, n : X_{2} → ℤ are defined by

F ∘ φ_{1}(x) = φ_{2}^{n(x)}(F (x)),
F ∘ φ_{1}^{m(x)}(x) = φ_{2}(F (x)),

then n, m are continuous on X \ {x_{0}} for some x_{0} ∈ X.

The countable group associated to (φ, X) itself arises as a homomorphic image of C(X, ℤ), the space of all continuous maps from X to (ℤ, +) with the product topology. As important as the details of the construction may be it is also remarkable that there is any reasonable way to assign a countable structure as a complete invariant.

Example 0.8. Stone spaces Perhaps this preceding example is reminiscent of the duality theorem of [75] for compact separable zero-dimensional Hausdorff spaces. To each such space we can assign a countable Boolean algebra with two spaces homeomorphic if and only if there exists an algebraic isomorphism between the Boolean algebras. Similarly Pontryagin duality, as it is found in [31], allows us to completely classify compact abelian metric groups by their countable discrete dual groups.

While it is true that complete invariants are provided this is not to say that these dualities are only or even primarily theorems of classification.

Example 0.9. Hom^{+}([0, 1]) Let Hom^{+}([0, 1]) be the group of all orientation preserving (π(0) = 0, π(1) = 1) homeomorphisms of the unit interval. The natural equivalence relation is that of conjugation: Two homeomorphisms, π_{1}, π_{2} are equivalent if a homeomorphic “relabeling” of the underlying space transforms one to the other, so that there is some σ ∈ Hom^{+}([0, 1]) with

π_{1} = σ^{−1} ∘ π_{2} ∘ σ.

Parallel to 0.3, the equivalence relation arises by the self-action of Hom^{+}([0, 1]) through conjugation.

It is sometimes felt that homeomorphisms of the unit interval are completely understood since we may represent each transformation symbolically by indicating the maximal regions on which we have either π(x) > x, π(x) = x, or π(x) < x. This can be made more precise by providing a classification of elements of Hom^{+}([0, 1]) by countable models. We naturally assign to each π ∈ Hom^{+}([0, 1]) a countable model M(π) such that for all π_{1}, π_{2} ∈ Hom^{+}([0, 1])

∃σ ∈ Hom^{+}([0, 1])(σ ∘ π_{1} ∘ σ^{−1} = π_{2}) ⇔ M(π_{1}) ≅ M(π_{2}).

The model M(π) consists of the maximal open intervals on which π displays one of the three possible behaviors indicated above. The language of M(π) encodes the linear ordering between these intervals and indicates which of the three possibilities hold. We will have an ordering ≤, and predicates P_{−}, P_{+}, and P_{=}. For I_{1} = (a_{1}, b_{1}), I_{2} = (a_{2}, b_{2}) maximal open intervals on which the behavior of π is unvarying, we have:

I_{1} ≤^{M(π)} I_{2} ⇔ a_{1} ≤ a_{2};
P_{−}^{M(π)}(I_{1}) ⇔ ∀x ∈ I_{1}(π(x) < x);
P_{+}^{M(π)}(I_{1}) ⇔ ∀x ∈ I_{1}(π(x) > x);
P_{=}^{M(π)}(I_{1}) ⇔ ∀x ∈ I_{1}(π(x) = x).

The outcome is similar for the homeomorphism group of Cantor space – ({0, 1})^{ℕ} – in the product topology. Since the homeomorphism group of the Cantor space is isomorphic to a closed subgroup of the infinite symmetric group it follows from [4] (see 2.39 below) that we may classify these homeomorphisms by countable models.

This is more than enough examples to be impressed by the diversity. But despite the variation, there are some common themes.

In the above we have natural numbers, real numbers, countable sets of complex numbers, countable ordinals, countable sets of countable ordinals, and various kinds of countable structures considered up to isomorphism being used as complete invariants. The connection between these examples is that in every one we may take a countable structure as a complete invariant; as in §2.3 below, we may code complex numbers, countable sets of complex numbers, countable ordinals, and countable sets of countable ordinals by appropriately chosen models. This suggests a notion of classification found at the opposing end of the spectrum to that of 0.1 and which is extreme in its generosity.

Question 0.10. Let E be an equivalence relation on a space X. When can we assign countable models or structures considered up to isomorphism as complete invariants?

Recall that HC is the collection of all hereditarily countable sets, and may be defined as the smallest collection of sets containing the natural numbers and closed under the operation of taking a countable subset. These therefore include all countable subsets of ℵ_{1}, all countable sets of subsets of ℕ, and – appropriately understood – all real numbers, all countable sets of real numbers, and so on.

In virtue of the Scott analysis of [59] we may equivalently ask:

Question 0.11. For which equivalence relations can we assign elements of HC as complete invariants?

[4] allows one more reworking of the question:

Question 0.12. Let E be an equivalence relation on a space X. When can we find a Polish space Y on which the infinite symmetric group S_{∞} acts continuously and a reasonable function θ : X → Y so that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ ∃g ∈ S_{∞}(g · θ(x_{1}) = θ(x_{2}))?

Of course we clearly need to have an assumption that the function θ or the assignments of models or HC sets be reasonable. On the whole I will take reasonable to mean Borel in an appropriate Borel structure, the technicalities of which are addressed in §2.1, §2.2, §2.3, and §3.1.^{1} But I should stress that relatively little change occurs if we extend to much broader classes of functions and far more generous methods of reduction. A point made in the course of §6.2 and §9.1-2 is that if there is any remotely definable assignment of countable models or HC sets in the context of Polish group actions, then we may find a reduction that is at worst only slightly more complicated than Borel.

This monograph can be viewed as part of a broad project to understand effective cardinalities in the sense raised by Luzin in [62], in the sense which reappears briefly in the opening parts of [11], but which finds its most forthright statement in modern works of descriptive set theory such as [60] and [50]. Here the concept is to calculate cardinalities using only functions that lay some claim on being reasonable or definable. Formally ℝ and ℝ/ℚ both have cardinality 2^{ℵ_{0}}; a well ordering of ℝ will enable us to find a bijection. Effectively ℝ is smaller than ℝ/ℚ, since we may find reasonable injections of ℝ into ℝ/ℚ (2.59, 2.63 below), but not the converse (3.8).

1In the context of Borel reduction, 0.10 and 0.12 are known to be equivalent problems from 2.7.3 of [4]. Provided we are willing to countenance reductions somewhat more complicated than Borel the Scott analysis of [59] shows both equivalent to 0.11. §6.1 and §9.1 return to these points.

Papers such as [14], [63], and [25] have previously addressed the question of which naturally occuring objects have effective cardinality no greater than that of ℝ. In a great many specific instances the answer has been determined, and one finds in [30] and [14] a kind of theory, recounted in §3.1 and §7.1, regarding when a reduction exists and why in certain cases it cannot. In turn this monograph tries to understand which objects have effective cardinality below HC and develop a parallel theory of why some do not.

We will only be concerned with the case that E arises from a Polish group action. Admittedly this may seem very restrictive, and it would certainly be desirable to have an analysis for all Borel or even Σ ∼_{1}^{1} equivalence relations. On the other hand most naturally occurring examples can be subsumed under an appropriately chosen Polish group action, and the impression left by chapter 8 is that the exceptions are somehow pathological. This is the point of question 10.9 near the end of the book, and even if that conjecture should fail it seems plausible that a similar outlook is justified.

§3 isolates a dynamical property for analyzing which Polish group actions allow reduction to countable models:

Definition 0.13. Let G be a topological group acting on a space X. The action is said to be turbulent if:

(i) every orbit is dense; (ii) every orbit is meager; (iii) for all x, y ∈ X, U ⊂ X, V ⊂ G open with x ∈ U, 1 ∈ V, there exists

y_{0} ∈ [y]_{G} =_{df} G · y and (g_{i})_{i∈ℕ} ⊂ V, (x_{i})_{i∈ℕ} ⊂ U with

x_{0} = x,
x_{i+1} = g_{i} · x_{i},

and for some subsequence (x_{n(i)})_{i∈ℕ} ⊂ (x_{i})_{i∈ℕ}

x_{n(i)} → y_{0}.

Turbulence is a sufficient condition for the orbit equivalence relation of a Polish group to refuse classification by countable structures; further: for a turbulent orbit equivalence relation any function assigning countable models up to isomorphism as invariants must be constant on a comeager set. (3.18, 3.19)

A transfinite analysis shows turbulence to be necessary for non-classification:

Theorem 0.14. Let G be a Polish group acting continuously on a Polish space X. Exactly one of the following holds:

1. the orbit equivalence relation E_{G}^{X} is reasonably reducible to isomorphism on countable models;

2. there is a turbulent Polish G-space Y and a continuous G-embedding from Y to X.

G V  U  X g_{0} g_{1}  g_{2} 1_{G} x_{0}  x_{1}  x_{2} x_{3} x_{i}  y_{0}  y x_{i+1} g_{i}

Figure 0.1. Turbulence

Here the notion of reduction is somewhat more complicated than Borel. In special cases, such as for G abelian or invariantly metrizable, one can obtain a reduction in 1. that is not only Borel but admits a Borel inverse up to orbit equivalence (6.40 and 6.30). The proof that 2. implies the negation of 1. is given in §3.2; the converse requires a long argument, finally concluding in chapter 9, and uses an elaboration on the Scott analysis presented in §6.2 that may have independent interest.

The form of 0.14 is intended to make it into a tool that can be easily applied in concrete cases.

xviii PREFACE

CHAPTER 1 An outline

This chapter discusses a few specific classification problems, but leaves until later the precise definitions.

1.1. Some specific classification problems

In light of 0.2, we may very well ask regarding the difficulty of classifying unitary operators on infinite dimensional Hilbert space. Let

l^{2} = {(x_{i})_{i∈ℕ}|x_{i} ∈ ℂ, ∑ |x_{i}|^{2} < ∞}

with the usual inner product and let U_{∞} be the set

{T : l^{2} → l^{2}|T linear, ∀x⃗, y⃗ ∈ l^{2}(⟨x⃗, y⃗⟩ = ⟨T (x⃗), T (y⃗)⟩)}.

We can set

T_{1} ≈ T_{2} ⇔ ∃S ∈ U_{∞}(ST_{1}S^{−1} = T_{2})

and obtain an infinite dimensional analog of the classification problem for unitary matrices. It was known from [10] that ≈ is non-smooth, in the sense of not allowing points in ℝ or any other Polish space as complete invariants.

Building on earlier results of Kechris’ the classifiability of ≈ by countable structures has been refuted in the following dramatic form:

Theorem 1.1. (Kechris, Sofronidis) The self action of U_{∞} by conjugation is turbulent (on an invariant dense G_{δ}).

While a satisfactory system of complete invariants have been established for the class of discrete spectrum measure preserving transformations of the unit interval and Bernoulli shifts, there is no known classification theorem for arbitrary measure preserving transformations. Certainly we might ask for classification by countable models or even by countable sets of reals.

It can be shown that we may reduce a turbulent orbit equivalence and conclude:

Theorem 1.2. (See [34]) The isomorphism relation on measure preserving transformations does not allow classification by countable models.

The classification theorem for locally compact groups from 0.4 hints at the existence of classifiability for broader classes of Polish groups actions, in particular those with a kinship to locally compact. Even though an infinite product group such as ℝ^{ℕ} would seem to bear little topological or algebraic kinship to S_{∞}:

Theorem 1.3. (§7.4) Let ∏_{ℕ} H_{n} be the product of countably many locally compact Polish groups and suppose it acts continuously on a Polish space. Then the resulting orbit equivalence relation allows classification by countable models.

Given the existence of a classification for orientation preserving homeomorphisms of the unit interval (0.9 and §4.2), we might hope for a similar classification of the homeomorphisms of the unit square. Here there is a distinction between dimensions 1 and 2.

Theorem 1.4. (§4.3) Let H = Hom([0, 1]^{2}) act upon itself by conjugation. Then the resulting equivalence relation does not admit classification by countable models.

After showing that the equivalence relation of conformal equivalence for Riemann surfaces is not reducible to the equality relation on any reasonably concrete space, the authors of [5] ask whether a classification may be produced using countable unordered sets of reals as complete invariants. [40] uses the uniformization theorem to reduce this classification problem to the orbit equivalence relation of a locally compact group, and then appeals to Kechris’ theorem as mentioned at 0.4 to obtain an affirmative answer.

No such result can be obtained for general complex manifolds:

Theorem 1.5. (Hjorth, Kechris; see [40]) Biholomorphism of 2-dimensional complex manifolds reduces a turbulent orbit equivalence relation; thus it does not admit classification by countable models, and in particular there is no reasonable method of assigning countable sets of reals as complete invariants.

Finally some words about the study of infinite dimensional group representations, since this is the area in which the smooth/non-smooth division received perhaps its first clear formulation. Given a countable group G we can consider the space of all representations on Hilbert space – that is to say, all homomorphisms of G into the infinite dimensional unitary group U_{∞}. The natural notion of classification is up to pointwise conjugation. Given ρ_{1}, ρ_{2} : G → U_{∞} let us write ρ_{1} ∼_{r} ρ_{2} if there is some T ∈ U_{∞} such that at each g ∈ G

T ∘ ρ_{1}(g) ∘ T^{−1} = ρ_{2}(g).

Already the Kechris-Sofronidis result shows that the classification problem for G = ℤ does not allow reduction to countable models, but one might argue that here we should restrict attention to the irreducible transformations. Building on [79]:

Theorem 1.6. ([35]) The irreducible representations considered up to ∼_{r} admit classification by countable models if and only if G is abelian-by-finite.

In §5 we prove this just for the specific case of G being the free group on infinitely many generators.

1.2. The form of this book (and one time paper)

This began as some seminars given at Caltech. The paper grew longer and longer, and at some point there entered the optimistic idea that by adding just a couple more pages it could be made totally self contained.

The desire to avoid any black boxes and reach a completely general audience has determined the choice of material. I have not tried to cover everything, but concentrated only on developing what seemed like the main ideas. In the exercises of §6.1 a set theorist may recognize that some calculations can be given swifter proofs using the relatively sophisticated techniques of [52] and [67]. §3.2 could be completed more quickly using “forcing”, but I have instead taken every step by hand.

Chapter 2 gives some background in classical descriptive set theory and equivalence relations. 3 provides the sufficient condition for non-classification:

Theorem 1.7. (See 3.19) A turbulent orbit equivalence relation does not allow classification by countable models.

and 4 and 5 applies this to homeomorphism groups and the unitary group of Hilbert space. Returning to the theory at 6 it is shown that turbulence is essential to nonclassification for a broad class of groups:

Theorem 1.8. Let G be a GE group (for instance abelian) and X a Polish G-space. Then the following are equivalent:

1. the orbit equivalence relation admits classification by countable models;

2. there is no turbulent Polish G-space Y and continuous G-embedding from Y to X.

In 7 a complete and self contained account is given of the GE groups and the Glimm-Effros dichotomies, and more generally the smooth\non-smooth dichotomy. There is a new result to the effect that countable products of locally compact groups admit a Glimm-Effros dichotomy. This in turn is used in the proof 1.3 at 7.43.

8 passingly covers Borel groups.

9 presents the general classifiability versus turbulence analysis, and is unlike the other chapters in the sense of being written purely with the expert in mind. I regret that it seemed impossible to give a self contained exposition of these proofs.

10 lists some open problems.

There is an appendix discussing ordinals, another clarifying notation, especially conventions that are idiosyncratic among set theorists, and there is an index.

Although the direction of this book is towards explicating a notion of classification there are also comments along the way regarding the broader area of Polish group actions. General results are not presented axiomatically in layered fashion, but are mostly developed at the point of use. Even as late as §7.1 basic facts appear for the first time.

The logical order of the chapters is that 2 precedes 3, and from there one can read any of chapters 4 through 8 just assuming those two. Chapter 9 however requires the reader to be familiar with most of the book.

For many purposes, shorter routes may be possible. Simply reading chapter §2.1, §2.2, and then up through the first half of §2.4, and §3.1, and §7.1-3 and §7.5 covers most of what is known about the Glimm-Effros dichotomy for Polish group actions. In a 10 week course at UCLA we went up through to 2.51, then §3.1, then lemma 3.14, which constitutes the essence of why turbulent equivalence relations are complicated, a few examples, and then finally we discussed §6.1.

There are many exercises, some of which are difficult and some of which are even theorems I wanted to mention briefly without taking a great deal of time to prove.

1.3. Acknowledgments

The work here was partially supported by NSF grant DMS 96-22977 and a generous grant from the Alfred P. Sloan Foundation.

2. Basic techniques

[figure] 8. Non-Polish groups

4. Homeomorphism groups of compact metric spaces

6.4 Another variation 3. Smoothness and turbulence

[figure]

[figure]

[figure]

[figure] [figure]

[figure] [figure]

[figure]

[figure]

[figure]

[figure] 5. Infinite dimensional representations

6.1 Scott’s analysis [figure]

[figure] 7.1 Glimm-Effros

[figure] [figure] 7.2 Solecki’s theorem. Nilpotent groups

6.2-3 A variation

[figure] 7.3 Changing topologies

[figure]

[figure]

[figure]

[figure] 7.4 Products of locally compact

7.5 Becker’s theorem

9. The analysis of turbulence

Figure 1.1. The order of dependence

I am indebted to Harvey Friedman for showing me the unpublished proof alluded to in [20].

I must twice thank Alexander Kechris.

Once for his general works on Polish groups, and many conversations, freely given, that served to introduce me to this area.

And once more for some very specific discussions on the notion of turbulence. During the Caltech seminars the definitions were the focus of his unrelenting criticism; this in turn molded the order of presentation at §3.2, and the decisions of where to make a theorem from a definition and where to do the reverse.

I am further grateful to Alexander Andretta, Lyman Chaffee, David Fremlin, Su Gao, and Nathan Schroeder for reading through various parts of earlier drafts and making invaluable comments. Undoubtedly obscurities and oversights remain, but they would number far greater without the suggestions I received. The research underpinning chapter 4 was especially influenced by a number of conversations with Su.

CHAPTER 2 Definitions and technicalities

2.1. Polish groups and Polish spaces

Definition 2.1. A topological space is said to be Polish if it is separable and admits a complete metric. When it becomes necessary we can write (X, τ) for a Polish space with underlying set X and topology τ. Mostly there is no danger of confusion and no harm in identifying a space with its underlying set. We will say that a metric d on a Polish space (X, τ) is compatible if it generates the topology; there is in particular no assumption of completeness.

For X a Polish space we form the Borel sets by taking the smallest collection containing the open sets and closed under complements and countable unions and intersections. In particular a countable intersection of open sets in X is G_{δ}, a countable union of closed is F_{σ}, countable intersections of F_{σ} are F_{σδ}, countable unions of G_{δ} are G_{δσ}, and so on. (This moves through the hierarchy of countable ordinals, with Σ ∼_{1}^{0} = open, Π ∼_{1}^{0} = closed, Σ ∼_{α}^{0} = countable unions of sets in Π ∼_{β}^{0} for various β < α, and Π ∼_{α}^{0} indicating the complements of Σ ∼_{α}^{0}.) A function θ : X → Y between Polish spaces is said to be Borel if the preimage of any open set is Borel.

For the purposes of verifying that certain maps and sets are Borel we will frequently be more interested in the fact that a space can be made Polish, and less interested in the details of how this is actually performed. If X is a set and B is a σ-algebra of subsets of X then one says that (X, B) is a standard Borel space if there is some Polish topology τ on X so that B is the collection of Borel sets for (X, τ).

Examples 2.2. Examples of Polish spaces include: ℝ with the usual topology; ℕ =_{df} {0, 1, 2, ...} with the discrete topology; the unit interval; the space of all real valued continuous functions from the unit interval with the sup norm –

d(f, g) = sup_{x∈[0,1]}|f(x) − g(x)|

(compare §4.1 below). Note that the class of Polish spaces is closed under countable products, since for X_{0}, X_{1}, ...X_{i}, ... Polish, as witnessed by metrics d_{0}, d_{1}, ...d_{i}, ..., we can find equivalent bounded metrics

d′_{i}(·, ·) = d_{i}(·, ·)/(d_{i}(·, ·) + 1)

and take

d_{Π}(x⃗, y⃗) = ∑ 2^{−i}d′_{i}(x_{i}, y_{i})

for x⃗, y⃗ in ∏ X_{i}. In particular any countable product of countable discrete spaces is Polish. For instance let ℕ^{ℕ} denote

{f| f : ℕ → ℕ}

with the product topology. (Similarly for n ∈ ℕ let ℕ^{n} be the set of functions from {0, 1, ..., n − 1} to ℕ, and let ℕ^{<ℕ} = ⋃ ℕ^{n}; one ruthlessly considers ∅, the empty function, as the unique element of ℕ^{0}.) Then we obtain that ℕ^{ℕ}, sometimes known as Baire space, is Polish.

A compact Polish space is presented by Cantor space:

2^{ℕ} =_{df} {f| f : {0, 1, 2, ...} → {0, 1}},

in the product topology, where we understand that 2 =_{df} {0, 1} is given the discrete topology and 2^{ℕ} is the collection of functions from ℕ to {0, 1}. (And for n ∈ ℕ we will use 2^{n} to denote the set of functions from {0, 1, ...n − 1} to {0, 1}, and 2^{<ℕ} for the union over n ∈ ℕ of 2^{n}). It is natural in turn to identify 2^{ℕ} with the collection of all subsets of ℕ – viewing each element f ∈ 2^{ℕ} as being married to the subset X_{f} of ℕ that has

n ∈ X_{f} ⇔ f(n) = 1.

A more exotic example of a Polish space is given by M(λ), the measurable subsets of [0, 1] considered up to null sets, so sets agreeing almost everywhere are identified. The metric here is

d(A, B) = λ(AΔB),

where AΔB is the set of points lying in one of the sets but not the other and where λ is Lebesgue measure. This is a complete metric, since if (A_{i})_{i∈ℕ} is Cauchy then by accelerating the sequence we can assume

λ(A_{i}ΔA_{i+1}) < 2^{−i},

from which it follows that taking

A_{∞} = {x ∈ [0, 1] : ∀i ∈ ℕ ∃j > i(x ∈ A_{j})}

will be a limit.

For our purposes there is an especially important example of a Polish space, namely the space of all L-structures on ℕ, for a countable language L. This example is the subject of §2.3.

Lemma 2.3. (classical) Let (X, τ) be a Polish space.

(i) If B ⊂ X is G_{δ} with respect to τ, then B is a Polish space in the relative topology.

(ii) If C ⊂ X is τ-closed and O ⊂ X is τ-open, then there is a Polish topology τ_{C,O} on X with basis

{C ∩ O ∩ U : U ∈ τ} ∪ {U : U ∈ τ}.

(iii) If τ is a Polish topology on X and for each i ∈ ℕ the topology τ_{i} ⊃ τ is a richer Polish topology, then the topology generated by taking finite intersections of sets from ⋃_{i∈ℕ} τ_{i} is Polish.

(iv) If B ⊂ X is Borel in τ then there is a richer Polish topology τ̂ ⊃ τ all of whose open sets are Borel with respect to τ and in which B is clopen, that is to say both τ̂-open and τ̂-closed.

Proof. (i): First consider the case of a single open set O ⊂ X. Choose closed C_{0} ⊂ ...C_{n} ⊂ C_{n+1} ⊂ ... with each C_{n} included in the interior of C_{n+1} and O = ⋃_{n∈ℕ} C_{n}. Let d be a complete metric on X bounded by 1. For a closed set C and x ∈ X let d(x, C) = inf{d(x, y) : y ∈ C}. Let X \ O indicate the points in X not appearing in O.

For each n, define f_{n}(x) to be

\frac{d(x, C_{n})/d(x, X \ O)}{[d(x, C_{n})/d(x, X \ O)] + 1},

where in the case d(x, X \ O) = 0 (and thus x ∈ X \ O) we let f_{n}(x) = 1. Thus we have a continuous function

f_{n} : X → [0, 1]

with f_{n} constantly 0 on C_{n} and f_{n} constantly 1 on X \ O.

Then

d_{O}(x_{0}, x_{1}) = d(x_{0}, x_{1}) + ∑_{n∈ℕ} |f_{n}(x_{0}) − f_{n}(x_{1})|

is a complete metric on O compatible with its topology as a subspace.

For the general case of a countable intersection of open sets and A = ⋂_{i} O_{i} we may take

d_{A}(·, ·) = ∑_{i} 2^{−i}d_{O_{i}}(·, ·)/(1 + d_{O_{i}}(·, ·)).

(ii): Write

O = ⋃_{n∈ℕ} C_{n},

with each C_{n} closed and included in the interior of C_{n+1}. We let f_{n} : X → [0, 1] be a continuous function with

f_{n}^{−1}[{1}] ⊃ C_{n},
f_{n}^{−1}[{0}] ⊃ X \ O.

Then define f : X → [0, 1] by

f(x) = ∑_{n∈ℕ} 2^{−n−1}f_{n}(x).

f is continuous and constantly 0 off of O.

Let X_{C,O} be the set of (x, r) ∈ X × [0, 1] such that either

x ∉ C, r = 0,

or

x ∈ C, r = f(x).

Here we have twisted X and immersed it into X × [0, 1]. (X \ C) ∪ (X \ O) is taped down to the bottom of the unit interval and C ∩ O pulled up though (0, 1], but with the end points, obtained as x → X \ O, trailing back to X × {0}.

Claim: X_{C,O} is a G_{δ} subset of X × [0, 1].

Proof of Claim: We have here that

X_{C,O} = (A_{00} ∩ A_{01}) ∪ (A_{10} ∩ A_{11}),

with each A_{ij} either closed or open, and therefore G_{δ}. Since the intersection of two G_{δ} sets are immediately G_{δ}, it suffices to see that the union of two G_{δ} sets is G_{δ}.

But given intersections of open sets

⋂_{n∈ℕ} O_{n}

and

⋂_{n∈ℕ} U_{n}

we can represent their unions as

⋂_{n∈ℕ}(( ⋂_{k<n} O_{n}) ∪ ( ⋂_{k<n} U_{n})).

(Claim□)

Thus by part (i), X_{C,O} is Polish. We moreover have a canonical bijection

π : X ↠ X_{C,O},
x ↦ (x, f(x))

if x ∈ C,

x ↦ (x, 0)

if x ∉ C. We can therefore let τ_{C,O} be the Polish topology on X obtained by sets of the form

π^{−1}[O],

where O ⊂ X_{C,O} is open.

This clearly includes the original topology, and without difficulty can be seen to have the basis indicated by the statement of (ii).

(iii): For τ_{∞} the topology thus generated in the statement of (iii) the space (X, τ_{∞}) is homeomorphic to

{x⃗ ∈ ∏_{i∈ℕ}(X, τ_{i}) : ∀n, m ∈ ℕ(x_{n} = x_{m})},

which in turn is a closed subspace of a Polish space.

(iv): The class of B ⊂ X satisfying the conclusion of (iv) includes the open sets by (ii), is closed under complementation by the structure of the definition, and is closed under countable unions by (iii) and (ii) again. So it forms a σ-algebra including the open sets, and thus the Borel.

In particular from (iv) we have that the class of standard Borel spaces is closed under taking Borel subsets.

Example 2.4. For X a Polish space and F(X) the space of closed subsets of X, we may find it natural to give F(X) the Borel structure generated by sets of the form {F ∈ F(X); F ∩ O ≠ ∅} for O ⊂ X open. This is known as the Effros Borel structure. F(X) equipped with the Effros Borel structure is a standard Borel space. One way to see this is to note that if we fix some countable basis B, and if we choose some complete metric d, and if we endow

2^{B} =_{df} {f| f : B → {0, 1}}

with the product topology – under which it is not just Polish but homeomorphic to Cantor space – then we may identify F(X) with the G_{δ} subset X_{0} of 2^{B} consisting of all x : B → {0, 1} such that

(i) U ⊂ V and x(U) = 1 implies x(V) = 1;

(ii) for all n ∈ ℕ, U ∈ B, if x(U) = 1 then there is some V ∈ B with

x(V) = 1,
d(V)(=_{df} sup{d(y, z) : y, z ∈ V }) < 2^{−n},
V ≠ ∅,

and the closure of V is included in U.

So for C ∈ F(X) we define x_{C} : B → {0, 1} by the requirement x_{C}(V) = 1 if and only if C ∩ V ≠ ∅.

Claim: C ↦ x_{C} defines a bijection between F(X) and X_{0}.

Proof of claim: The map is clearly one to one, so there is only the task of showing it onto. Take x ∈ X_{0} and define C ⊂ X by

C = X \ (⋃{V : x(V) = 0}).

We need to show that x = x_{C}.

Clearly x(V) = 0 implies V ∩C = ∅, and hence x_{C}(V) = 0. On the other hand, if x(V) = 1 we may use clause (ii) in the definition of X_{0} to find V_{0} = V ⊃ V_{1} ⊃ V_{2}... ⊃ V_{i}... such that at each i we have x(V_{i}) = 1, d(V_{i}) < 2^{−i}, and the closure of V_{i+1} included in V_{i}. Letting z be the unique point in

⋂_{n∈ℕ} V_{n}

it suffices to see that z ∈ C. But otherwise we would obtain some open U ∈ B with x(U) = 0 and z ∈ U. From the construction we obtain V_{n} ⊂ U for some n, and a contradiction to clause (i) in the definition of X_{0}. (Claim□)

The claim implies that X_{0} equipped with its Borel σ-algebra is isomorphic to F(X) equipped with the Effros σ-algebra.

Definition 2.5. A Polish group is a topological group that is Polish as a space. In the case that G is a Polish group acting continuously on a Polish space X we say that X is a Polish G-space; if X is a standard Borel space and G acts in a Borel fashion, then X is said to be a Borel G-space. In either case we write E_{G}^{X} – or E_{G} when context indicates X – for the orbit equivalence relation given by

x_{1}E_{G}^{X}x_{2} ⇔ ∃g ∈ G(g · x_{1} = x_{2}).

Here g · x denotes the result of g ∈ G acting on x ∈ X. The orbit of x is the set {g · x : g ∈ G} and denoted by [x]_{G}; thus x_{1}E_{G}^{X}x_{2} if and only if [x_{1}]_{G} = [x_{2}]_{G}. We let

G_{x} =_{df}= {g ∈ G : g · x = x}

be the stabilizer of x, consisting of group elements that leave x undisturbed.

For A ⊂ X

g · A = {g · x : x ∈ A};

A is said to be invariant if for all g ∈ G

g · A = A.

Given F ⊂ G and A ⊂ X we let F · A equal the set {g · x : g ∈ F, x ∈ A}. G · A =_{df} {g · x : g ∈ G, x ∈ A} is said to be the saturation of A.

Examples 2.6. Any countable discrete group is Polish. (ℝ, +) is a Polish group. ℝ^{ℕ} in the product group structure. Any separable Banach space; for instance if we let

c_{0} = {x⃗ ∈ ℝ^{ℕ} : x_{i} → 0},

then

d(x⃗, y⃗) = sup|x_{i} − y_{i}|

is a metric witnessing it Polish. GL_{2}(ℝ), along with the other Lie groups, is Polish. 2^{ℕ} may be viewed as a Polish group under the identification with (ℤ_{2})^{ℕ} given by thinking of ℤ_{2}, the abelian group of order 2, as having elements 0 and 1. M(λ), the measurable subsets of [0, 1], with the group operation of symmetric difference is a Polish group.

Definition 2.7. A group G along with an algebra B is said to be a Borel group if (G, B) is a standard Borel space and the group operation of inversion is measurable with respect to B and the group operation of multiplication for G is measurable with respect to the σ-algebra on G×G generated by B×B. Equivalently, if τ is any Polish topology on G generating B as its Borel sets,

(()^{−1})^{G} : (G, τ) → (G, τ),
g ↦ g^{−1},

and

(() · ())^{G} : (G, τ) × (G, τ) → (G, τ),
(g, h) ↦ g · h,

are Borel functions.

A Borel group (G, B) is said to be Polishable if there is some Polish topology τ giving rise to B as its collection of Borel sets and for which (G, τ) is a Polish group.

Examples 2.8. l^{1} = {x⃗ ∈ ℝ^{ℕ} : ∑_{i∈ℕ} |x_{i}| < ∞} is a standard Borel group under the operation of pointwise multiplication and in the Borel structure it obtains from being a Borel subset of ℝ^{ℕ}. The metric derived from the l^{1}-norm,

d(x⃗, y⃗) = ∑_{i∈ℕ} |x_{i} − y_{i}|,

shows it to be Polishable. c_{00} = {x⃗ ∈ ℝ^{ℕ} : ∃m∀i > m(x_{i} = 0)} is an example of a non-Polishable group to which chapter 8 will return.

Definition 2.9. Let λ denote the usual Lebesgue measure on [0, 1]. We then define L^{2}([0, 1], λ), or just L^{2} for short, to be the separable Hilbert space consisting of all square integrable complex valued functions considered up to λ-a.e. equality. Then we have the inner product

⟨f, g⟩ = ∫_{[0,1]} f · \overline{g} λ,

where here \overline{g} denotes the complex conjugate of g. Let || · ||_{2} denote the Hilbert space norm. Let U_{∞} be the unitary group of L^{2}, consisting of all continuous linear operators that respect the inner product; we give this group the operator topology, so that a subbasic open set will have the form

{T ∈ U_{∞} : ||T (e) − T_{0}(e)||_{2} < ϵ}

for some T_{0} ∈ U_{∞}, ϵ > 0, e ∈ L^{2}.

Let M_{∞} be the group of invertible measure preserving transformations of the unit interval where we identify two transformations that agree a.e. M_{∞} is given the topology arising from viewing it as a subgroup of the unitary group on L^{2}([0, 1], λ). To each measure preserving bijection π : [0, 1] → [0, 1] we can associate

T_{π} : L^{2}([0, 1], λ) → L^{2}([0, 1], λ)

by

T_{π}(f) = f ∘ π^{−1}.

Finally, let S_{∞}, or Sym(ℕ), be the infinite symmetric group on ℕ consisting of all permutations of ℕ with the topology of pointwise convergence. Thus a basic open set will have the form

{f ∈ S_{∞} : f(n_{1}) = k_{1}, ...f(n_{l}) = k_{l}}

for some sequence n_{1}, n_{2}, ..., n_{l}, k_{1}, ..., k_{l}.

S_{∞} in the above topology is a G_{δ} subset of ℕ^{ℕ} and hence Polish by 2.3. The proofs for the other two groups are not quite so quick.

Lemma 2.10. U_{∞} is a Polish group.

Proof. Let d(·, ·) be the bounded complete metric on L^{2} given by

d(e, e′) = \frac{||e − e′||_{2}}{(||e − e′||_{2} + 1)}.

Let (e_{i})_{i∈ℕ} be a dense subset of the Hilbert space.

Then a metric on U_{∞} is formed by

d^{∗}(T, S) = ∑ 2^{−i}(d(T (e_{i}), S(e_{i})) + d(T^{−1}(e_{i}), S^{−1}(e_{i}))).

This is easily seen to give a metric compatible with the topology, since if

U = {T ∈ U_{∞} : ||T (e) − T_{0}(e)||_{2} < ϵ}

and

T ∈ U,

then we may find some δ > 0 with

||T (e) − T_{0}(e)||_{2} < ϵ − 3δ

and some i ∈ ℕ with

||e − e_{i}||_{2} < δ.

Then for any other S ∈ U_{∞} with

d^{∗}(S, T) < 2^{−i}(δ/(δ + 1))

we obtain

||S(e_{i}) − T (e_{i})||_{2} < δ

and hence

||S(e) − T (e)||_{2} < 3δ

since both transformations are isometries of L^{2}. Hence

||S(e) − T_{0}(e)||_{2} < ϵ

by the triangle inequality.

As for completeness of the metric, note that if (T_{j})_{j∈ℕ} is d^{∗} Cauchy, then we obtain two linear isometries S and T from the sets {e_{i} : i ∈ ℕ} to L^{2} such that at each i

T_{j}(e_{i}) → T (e_{i})

and

T_{j}^{−1}(e_{i}) → S(e_{i})

as j → ∞. Since T and S are isometries defined on dense subsets of L^{2} they extend to linear isometries T̂, Ŝ : L^{2} → L^{2}, which we then obtain to be isomorphisms of the Hilbert space by observing that T̂^{−1} = Ŝ.

Lemma 2.11. M_{∞} is a Polish group.

Proof. Fix {B_{n} : n ∈ ℕ} a countable algebra of subsets of [0,1] including the intervals with rational endpoints. Then at each n we can let χ_{B_{n}} : [0, 1] → {0, 1} be the characteristic function of B_{n},

χ_{B_{n}}^{−1}[{1}] = B_{n},
χ_{B_{n}}^{−1}[{0}] = [0, 1] \ B_{n},

where we choose to consider this function as an element of L^{2}([0, 1], λ). For π_{0}, π_{1} ∈ M_{∞} let

d(π_{0}, π_{1}) = ∑_{i∈ℕ} 2^{−i}[λ(π_{0}(B_{i})Δπ_{1}(B_{i})) + λ(π_{0}^{−1}(B_{i})Δπ_{1}^{−1}(B_{i}))].

Since the finite linear combinations of the functions {χ_{B_{n}} : n ∈ ℕ} are dense in L^{2}([0, 1], λ) we obtain that d is compatible with the topology on M_{∞}.

If (π_{i})_{i∈ℕ} is Cauchy then we can assume that for all j, j′ > l > i

λ(π_{j}(B_{i})Δπ_{j′}(B_{i})) + λ(π_{j}^{−1}(B_{i})Δπ_{j′}^{−1}(B_{i})) < 2^{−l}

and thus, as in the proof of the completeness of M(λ) from 2.2, obtain the needed A_{i} and C_{i} measurable subsets of [0, 1] such that

π_{j}(B_{i}) → A_{i}
π_{j}^{−1}(B_{i}) → C_{i}

as

j → ∞.

Note that the map B_{i} ↦ A_{i} defines an isometry on a dense subset of M(λ), the space of all measurable subset of [0,1]. Hence it extends to an isometry

π̂ : M(λ) → M(λ).

Since B_{i} ↦ C_{i} extends to an inverse for π̂ we obtain that it is onto. Since the Boolean operations are continuous, π̂ is actually an automorphism of the Boolean algebra of the measurable subsets of [0,1]. We will be finished if we can somehow produce π : [0, 1] → [0, 1] such that for all A ⊂ [0, 1] in M(λ) we have π[A] = π̂(A).

Let {I_{s} : s ∈ 2^{<ℕ}} be a family of closed intervals such that at each n ∈ ℕ (i) d(I_{s}) = 2^{−n} for each s : {0, 1, ..., n − 1} → {0, 1}; (ii) for s, t : {0, 1, ..., n − 1} → {0, 1} with s ≠ t, I_{s} ∩ I_{t} contains at most a single point; (iii) [0,1] equals

⋃_{s:{0,1,...,n−1}→{0,1}} I_{s}.

It follows from the assumption that π̂ defines an automorphism of the algebra of measurable sets that for λ a.e. x ∈ [0, 1] there will be a unique point in the intersection

⋂_{{s:x∈π̂^{−1}(I_{s})}} I_{s},

which we can then define to be π(x), the value of the transformation π at x. Letting B_{0} be the dense subset of M(λ) generated by {I_{s} : s ∈ 2^{<ℕ}} we have that for all A ∈ B_{0}

π^{−1}[A] = π̂^{−1}(A).

But since for any sequence (D_{i})_{i∈ℕ} of measurable sets

π^{−1}(⋃_{i∈ℕ} D_{i}) = ⋃_{i∈ℕ} π^{−1}(D_{i})

π^{−1}(⋂_{i∈ℕ} D_{i}) = ⋂_{i∈ℕ} π^{−1}(D_{i})

and since π̂^{−1} is an automorphism of the algebra of measurable sets, we obtain that π^{−1} and π̂^{−1} agree on the σ algebra generated by B_{0}, as required.

One can also establish 2.11 by showing M_{∞} to be a closed subgroup of U_{∞}.

Examples 2.12. A very simple kind of example of a Polish G-space is when G may be continuously injected into some other Polish group and we take the induced action by translation. c_{0} with the sup norm continuously injects into ℝ^{ℕ} with the usual product Polish structure; the orbit equivalence relation resulting from this action is the coset equivalence relation.

More interesting are the self actions of reasonably rich groups such as U_{∞}, M_{∞}, and S_{∞} by conjugation. So for σ and π in M_{∞} we have the action π·σ = π∘σ∘π^{−1}. The orbit equivalence relation resulting from this action of M_{∞} is the subject of [34]. The action of U_{∞} by self conjugation is analyzed in the course of the spectral theorem (see [83]). The action of S_{∞} on itself by conjugation is on the other hand trivial, and completely described by 3.11 below.

Exercise 2.13. Let G be a topological group.

(i) If is V an open neighborhood of the identity then there is an open neighborhood of the identity W with W^{2}(=_{df} {gh : g, h ∈ W }) ⊂ V and 1_{G} ∈ W and W^{−1}(=_{df} {g^{−1} : g ∈ W }) = W.

(ii) If H < G is an open subgroup then H is clopen.

(iii) If G acts continuously on a metric space X then for all x ∈ X the set

G_{x} =_{df} {g ∈ G : g · x = x}

forms a closed subgroup.

(iv) If Q ⊂ G is dense and U ⊂ G open then G = QU(=_{df} {g·h : g ∈ Q, h ∈ U}).

Exercise 2.14. Let G be a Polish group and X be Polish G-space.

(i) If O ⊂ X is open so is

{x ∈ X : ∃g ∈ G(g · x ∈ O)},

the set of all points whose orbits meet O.

(ii) If C ⊂ X is a dense G_{δ} and g ∈ G, then g · C is again a dense G_{δ}.

Exercise 2.15. (i) Let (g_{i})_{i∈ℕ} be a sequence of group elements in S_{∞} such that for all i, j ≥ l

g_{i}(l) = g_{j}(l)
g_{i}^{−1}(l) = g_{j}^{−1}(l).

Show there is some limit g ∈ S_{∞}

g_{i} → g.

(ii) Show that simply assuming for all i, j ≥ l

g_{i}(l) = g_{j}(l)

does not suffice to establish a limit.

Exercise 2.16. Let X be Polish and suppose for each n we have B_{n} ⊂ X Borel.

(i) The set

{x ∈ X : ∃n ∈ ℕ(x ∈ B_{n})}

equals

⋃_{n∈ℕ} B_{n}

and is thus Borel.

(ii) The set

{x ∈ X : ∀n ∈ ℕ(x ∈ B_{n})}

equals

⋂_{n∈ℕ} B_{n}

and is thus Borel.

The catch phrase is that “Borel sets are closed under number quantifiers.” A similar result holds for the ∃^{∞} quantifier, where for x ∈ X write ∃^{∞}n(x ∈ B_{n}) if there are infinitely many n with x ∈ B_{n}.

(iii) Then the set of {x ∈ X : ∃^{∞}n(x ∈ B_{n})} is Borel.

(iv) If each B_{n} is open then {x ∈ X : ∃^{∞}n(x ∈ B_{n})} is G_{δ}.

(v) Let (f_{n})_{n∈ℕ} be a sequence of Borel functions from X to [0, 1]. Then the function

x ↦ lim inf_{n∈ℕ}f_{n}(x)

is again a Borel function.

Exercise 2.17. (i) If f : X → Y is a continuous function between Polish spaces, and B ⊂ Y is G_{δ} (or F_{σ}, G_{δσ}, F_{σδ}, and so on), then f^{−1}[B] is G_{δ} (or respectively F_{σ}, G_{δσ}, F_{σδ}).

(ii) If X, Y are Polish and B ⊂ X × Y is G_{δ} (or F_{σ}, G_{δσ}, F_{σδ}, and so on) in the product topology, then for any x ∈ X the set B_{x} = {y ∈ Y : (x, y) ∈ B} is G_{δ} (or respectively F_{σ}, G_{δσ}, F_{σδ}).

(iii) If f : X → Y is a Borel function between Polish spaces then there is a richer Polish topology in which f is continuous.

(iv) If f : X → Y is a Borel function between Polish spaces then for any Borel B ⊂ Y we have f^{−1}[B] to be a Borel subset of X.

Exercise 2.18. If f : X → Y is a Borel function between Polish spaces, then the graph of f,

{(x, y) ∈ X × Y : f(x) = y},

is Borel as a subset of X × Y.

Exercise 2.19. Let (X, d) be a complete metric space. Suppose we have nonempty open sets (U_{n})_{n∈ℕ} such that at each n

\overline{U_{n+1}} ⊂ U_{n},
d(U_{n}) =_{df} sup{d(x_{0}, x_{1}) : x_{0}, x_{1} ∈ U_{n}} < 2^{−n}.

Then there is a unique point

x ∈ ⋂_{n∈ℕ} U_{n}.

Exercise 2.20. Let X be a Polish space and suppose X_{0} ⊂ X is a subspace which is Polish in the induced subspace topology. Then X_{0} is G_{δ} in X.

2.2. Equivalence relations

Definition 2.21. For X Polish, an equivalence relation E ⊂ X × X is Borel if it is Borel with respect to the product topology on X × X. For E an equivalence relation on a space X and x ∈ X,

[x]_{E} =_{df} {z ∈ X : zEx}

indicates the equivalence class of x. We say that A ⊂ X is E-invariant if every E-equivalence class is either included in A or included in X \A (where here X \A is the complement of A). For E and F equivalence relations on X and Y we say that E is Borel reducible to F, written E ≤_{B} F, if there is a Borel function θ : X → Y so that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ θ(x_{1})F θ(x_{2}).

We write E ≤_{c} F if there is a continuous reduction, and E ⊑_{B} F and E ⊑_{c} F if the reduction may respectively chosen Borel and one to one or continuous and one to one. We can write E <_{B} F if E ≤_{B} F holds and F ≤_{B} E fails. E ≰_{B} F and E ≰_{c} F indicate that E is not Borel reducible, in the first case, or not to continuously reducible, in the second, to F. In the case that G acts on the Polish space space X we may as well write X/G ≤_{B} F and F ≤_{B} X/G for E_{G}^{X} ≤_{B} F and F ≤_{B} E_{G}^{X} respectively; similarly for group actions [x]_{G} stands in place of [x]_{E_{G}^{X}}.

Definition 2.22. E_{0} is the equivalence relation of eventual agreement on elements of 2^{ℕ}, so that

xE_{0}y ⇔ ∃m∀n > m(x(n) = y(n)).

E_{0} is a Borel equivalence relation, and in fact it is F_{σ} in 2^{ℕ} ×2^{ℕ}. For a Polish space X, we let id(X) denote the equality relation on X. For x, y ∈ 2^{ℕ×ℕ} we write xF_{2}y if

{x(n, ·) : n ∈ ℕ} = {y(n, ·) : n ∈ ℕ},

where x(n, ·) is the element of x_{n} of 2^{ℕ} defined by x_{n}(m) = x(n, m); so in other words, x and y are F_{2} equivalent if they code the same countable set of elements of Cantor space. Similarly for x, y ∈ 2^{ℕ×ℕ×ℕ} we write xF_{3}y if

{{x(n, l, ·) : n ∈ ℕ} : l ∈ ℕ} = {{y(n, l, ·) : n ∈ ℕ} : l ∈ ℕ}.

More generally we can iterate this definition, defining for x, y ∈ 2^{(ℕ^{k})} the equivalence relation xF_{k}y to hold if {[x(n, ·, ·, ...)]_{F_{k−1}} : n ∈ ℕ} = {[y(n, ·, ·, ...)]_{F_{k−1}} : n ∈ ℕ}. (And here the authors of [20] continue these definitions through all the countable ordinals, providing F_{α} for α < ω_{1}.) By induction on k each F_{k} can be shown to be Borel – for instance F_{2} is F_{σδ}. With a minor blurring of vision we may view these F_{k}’s as orbit equivalence relations induced by S_{∞}. For instance let B_{2} be the set of x ∈ 2^{ℕ×ℕ} such that for all m ≠ n

x(n, ·) ≠ x(m, ·)

and let S_{∞} act on B_{2} by

g · x(n, m) = x(g^{−1}(n), m).

Here we have that E_{S_{∞}}^{B_{2}} (the orbit equivalence relation from this action of S_{∞} on B_{2}) equals F_{2}|_{B_{2}} (the restriction of F_{2} to B_{2}) and

F_{2} ≤_{B} F_{2}|_{B_{2}} ≤_{B} F_{2}.

One direction of this is trivial; F_{2} ≤_{B} F_{2}|_{B_{2}} is left as an exercise below.

Lemma 2.23. (folklore) Let G be a countable discrete group and X a Borel G-space. Then E_{G}^{X} ≤_{B} F_{2}.

Proof. Rephrasing our assumptions, there is a countable group (f_{n})_{n∈ℕ} of Borel bijections such that for each x ∈ X

[x]_{G} =_{df} {y ∈ X : xE_{G}^{X}y} = {f_{n}(x) : n ∈ ℕ}.

Let {U_{l} : l ∈ ℕ} be a countable basis for X, the space on which E is defined. For x ∈ X let f̂(x) ∈ 2^{ℕ} be given by

(f̂(x))(l) = χ_{U_{l}}(x),

where χ_{U_{l}} is the characteristic function of U_{l}, taking the value 1 if x ∈ U_{l} and 0 otherwise. f̂: X → 2^{ℕ} is Borel.

θ : X → 2^{ℕ×ℕ}

defined by

(θ(x))(n, m) = (f̂∘ f_{n}(x))(m)

witnesses E_{G}^{X} ≤_{B} F_{2}.

Another class of group actions are those arising from shifting subsets of G or some collection of functions on a G-set.

Definition 2.24. If G is a Polish group then we can let it act by shift on the standard Borel space F(G) of all closed subsets of G by

h ∈ g · F ⇔ g^{−1}h ∈ F.

For G a countable group we have the shift action on 2^{G}(=_{df} {f |f : G → {0, 1}}) given by

(g · x⃗)(h) = x⃗(g^{−1}h)

where x⃗ ∈ 2^{G}, g, h ∈ G; note that in the product topological structure 2^{G} is a compact metrizable space and the action is continuous.

Lemma 2.25. (classical) id(ℝ) <_{B} E_{0} ≤_{B} ℝ/ℚ, 2^{ℤ}/ℤ.

Proof. The first strict inequality is in §3.1; E_{0} ≤_{B} 2^{ℤ}/ℤ and E_{0} ≤_{B} ℝ/ℚ follow from 7.9.

Definition 2.26. An equivalence relation E is said to be smooth if E ≤_{B} id(ℝ).

Exercise 2.27. (i) id(ℝ) ≤_{B} id(2^{ℕ}) ≤_{B} id(ℝ). (For instance: For r ∈ ℝ and (q_{n})_{n∈ℕ} an enumeration of ℚ define θ(r) ∈ 2^{ℕ} by (θ(r))(n) = 1 if and only if r < q_{n}. Conversely, to each x ∈ 2^{ℕ} we can associate the real r_{x} ∈ [0, 1] that has the value x(i) in the ith position of its decimal expansion.)

(ii) Let E be an equivalence relation on a Polish space X. E is smooth if and only if there is a countable sequence of (A_{n})_{n∈ℕ} of Borel invariant sets such that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ ∀n(x_{1} ∈ A_{n} ⇔ x_{2} ∈ A_{n}).

(iii) For E and X as in (ii), let B_{E} be the subsets of X/E generated by the Borel invariant sets, so that for each E-invariant Borel set B ⊂ X we will have the corresponding B/E = {[x]_{E} : x ∈ B} in B_{E}. If (X/E, B_{E}) is a standard Borel space then E is smooth.^{1}

Exercise 2.28. Show ℝ/ℤ ≤_{B} id(ℝ).

Exercise 2.29. (i) Define E_{0}(ℕ^{ℕ}) on ℕ^{ℕ} by analogy with E_{0}, so that for x, y ∈ ℕ^{ℕ} we set xE_{0}(ℕ^{ℕ})y if

∃m ∈ ℕ ∀n ≥ m(x(n) = y(n)).

Show E_{0}(ℕ^{ℕ}) ≤_{c} E_{0} ≤_{c} E_{0}(ℕ^{ℕ}).

(ii) More generally, if A is a countably infinite set and B is a countable or finite set of size at least 2, we may define E_{0}(B^{A}) by the specification that for x, y ∈ B^{A} we have xE_{0}(B^{A})y if there is some finite F ⊂ A such that

∀a ∈ A \ F (x(a) = y(a)).

Here again E_{0}(B^{A}) ≤_{c} E_{0} ≤_{c} E_{0}(B^{A}).

Exercise 2.30. Show that F_{2} ≤_{B} F_{2}|_{B_{2}}. (Hint: For m ∈ ℕ and x ∈ ℕ^{ℕ} we will define (0^{m}1)^{⌢}x by

1. ((0^{m}1)^{⌢}x)(k) = 0 for k < m;

2. ((0^{m}1)^{⌢}x)(k) = 1 for k = m;

3. ((0^{m}1)^{⌢}x)(k) = x(k − m − 1) for k > m.

It is then possible to find a Borel function x ↦ y_{x} such that for all x ∈ 2^{ℕ×ℕ} we have {y_{x}(n, ·) : n ∈ ℕ} = {((0^{m}1)^{⌢}x)(n, ·) : n ∈ ℕ} and y_{x} ∈ B_{2}.)

Exercise 2.31. Let F be an equivalence relation on a Polish space all of whose equivalence classes are countable. Show F_{2} ≰_{c} F as follows:

(i) Observe that B_{2} from 2.22 is a Polish S_{∞}-space and it suffices to show

E_{S_{∞}}^{B_{2}} ≰_{c} F.

(ii) For N ∈ ℕ let E_{N} be the set of g ∈ S_{∞} such that for all i ≤ N

g(i) = i.

If x ∈ B_{2} is such that for all s ∈ 2^{<ℕ} there exists n ∈ ℕ

x(n, ·) ⊃ s

then

\overline{E_{N} · x} = \{y ∈ B_{2} : ∀i ≤ N(x(i, ·) = y(i, ·))\}.

(Here \overline{A} indicates the topological closure of a set A.) (iii) If we have a continuous

θ : B_{2} → X

such that for all x ∈ B_{2} and y ∈ [x]_{S_{∞}}

θ(x)F θ(y)

then for each x ∈ B_{2} there exists some x_{0} ∈ [x]_{S_{∞}} and N ∈ ℕ so that θ(E_{N} · x_{0}) consists in a single point. (The Baire category theorem implies that S_{∞} is not a countable union of closed nowhere dense sets, and so we can find z_{0} ∈ [θ(x)]_{F} for which {g ∈ S_{∞} : θ(g · x) = z_{0}} contains an open set.)

1Although this implication does not in general reverse for arbitrary Borel equivalence relations, there is a converse for the cases of most interest to us here. Burgess in [8] shows that if a Polish G acts continuously on Polish X then E_{G}^{X} ≤_{B} id(ℝ) if and only if (X/E_{G}^{X}, B_{E_{G}^{X}}) is standard Borel.

(iv) Conclude that for x as in (ii) and x_{0} and N as in (iii) there exists some y ∉ [x]_{S_{∞}} with

y ∈ \overline{E_{N} · x_{0}}
∴ θ(x_{0}) = θ(y).

We will see at 2.64 that F_{2} is not even Borel reducible to a countable equivalence relation.

Exercise 2.32. For E a Borel equivalence relation on a Polish space X, the following three things are equivalent:

(i) E ≤_{B} F_{2};

(ii) There is a countable sequence (f_{n})_{n∈ℕ} of Borel functions from X to ℝ such that for all x_{0}, x_{1} ∈ X we have x_{0}Ex_{1} if and only if

{f_{n}(x_{0}) : n ∈ ℕ} = {f_{n}(x_{1}) : n ∈ ℕ}.

(Here we are only asking that {f_{n}(x_{0}) : n ∈ ℕ} and {f_{n}(x_{1}) : n ∈ ℕ} be equal as unordered sets.)

(iii) There is a countable sequence (f_{n})_{n∈ℕ} of Borel functions from X to 2^{ℕ} such that for all x_{0}, x_{1} ∈ X we have x_{0}Ex_{1} if and only if

{f_{n}(x_{0}) : n ∈ ℕ} = {f_{n}(x_{1}) : n ∈ ℕ}.

In general for a set A we let P_{ℵ_{0}}(A) indicate the countable subsets of A. In light of 2.32 it seems natural to say that 2^{ℕ×ℕ}/F_{2} has the same effective cardinality as P_{ℵ_{0}}(ℝ).

2.3. Spaces of countable structures

Intuitively the idea of a class of discrete structures should be clear. For instance, any abstract group (freed from any topology) is an example of a discrete structure. Similarly we can consider the class of all rings or fields, or any other notion of algebraic object. Of course we do not need to restrict ourselves to discrete structures appearing in abstract algebra. Other classes of structures are given by graphs and linear orderings.

The purpose of this particular section is to fix a precise definition of the space, Mod(L), of countable structures or countable models for a given language, L. This will provide clear meaning to the idea of a Borel map into this space and the notion of a Borel subset of this space. Most of the natural classes of countable structures, such as all those alluded to above, can be identified with a Borel subset of some appropriately chosen Mod(L).

Definition 2.33. Let L = {R_{i} : i ≤ k} be a finite relational language. So each R_{i} will be a relation with some fixed arity, a(i) – which is to say its interpretation in any L-structure will be as some relation on sequences of length a(i).

Then Mod(L) is the space of all L-structures with underlying set ℕ. Thus each M ∈ Mod(L) is a sequence of assignments: to each R_{i} we have a set

R_{i}^{M} ⊂ ℕ^{a(i)}.

In this respect M can be formally viewed as an element of

∏_{i≤k} P(ℕ^{a(i)}),

where each P(ℕ^{a(i)}) is the set of all subsets of ℕ^{a(i)}.

This space is equipped with a topology whose subbasic open sets have form

{M ∈ Mod(L) : (k_{1}, k_{2}, ..., k_{a(i)}) ∈ R_{i}^{M}}

or

{M ∈ Mod(L) : (k_{1}, k_{2}, ..., k_{a(i)}) ∉ R_{i}^{M}},

for some i ≤ k, k_{1}, k_{2}, ..., k_{a(i)} ∈ ℕ.

We can similarly define Mod(L) in the case that L = {R_{i} : i ∈ ℕ} is a countably infinite relational language, and obtain a natural identification with

∏_{i∈ℕ} P(ℕ^{a(i)}).

Examples 2.34. Thus if L = {R}, where R has arity 2, then every M ∈ Mod(L) is a directed graph on ℕ, and can be naturally thought of as an element of 2^{ℕ×ℕ}, by associating to each M the function

χ_{R^{M}} : ℕ × ℕ → {0, 1},
χ_{R^{M}}^{−1}[{1}] = R^{M}.

If L = {P_{0}, P_{1}}, where P_{0}, P_{1} have arity 1 – that is to say they are unary predicates – then each element of M ∈ Mod(L) consists of the assignment of subsets P_{0}^{M} and P_{1}^{M} of ℕ.

Definition 2.35. It is customary to write

M ⊧ R_{i}(k_{1}, k_{2}, ..., k_{a(i)})

in place of (k_{1}, k_{2}, ..., k_{a(i)}) ∈ R_{i}^{M}, and then to define M ⊧ φ(k_{1}, ...) for all formulas of L by induction on the complexity of φ:-

M ⊧ ∃xφ(k_{1}, ..., x)

if and only if for some l ∈ ℕ

M ⊧ φ(k_{1}, ..., l);

and then

M ⊧ φ ∧ ψ(k_{1}, ...)

if and only if

M ⊧ φ(k_{1}, ...)

and

M ⊧ ψ(k_{1}, ...);

finally

M ⊧ ¬φ(k_{1}, ...)

if and only if it is not the case that

M ⊧ φ(k_{1}, ...).

As well as these primitive logical connectives, there are also the usual composites defined from them:

M ⊧ φ ⇒ ψ(k_{1}, ...)

stands for

M ⊧ ¬(φ ∧ (¬ψ(k_{1}, ...));
φ ⇔ ψ

is short hand for φ ⇒ ψ ∧ ψ ⇒ φ; ∀ for ¬∃¬; φ ∨ ψ for ¬(¬φ ∧ ¬ψ). Thus for example if M is a structure on ℕ for the language generated by a single binary relation R, then

M ⊧ ∀a∃bR(a, b)

if and only if it really is the case that for all a ∈ ℕ there exists b ∈ ℕ with

M ⊧ R(a, b).

The equality symbol is usually given a special status. One writes

M ⊧ a = b

exactly when it is indeed the case that a = b, and therefore without reference to any individual properties of M; consequently one would not normally allow the symbol = as part of a language L.

A formula is said to be quantifier free if it is built up from the atomic formulas of the form x_{0} = x_{1} and R_{i}(x_{1}, ..., x_{a(i)}) using just the boolean connectives ¬ and ∧.

In the special case of R having arity 2 – that is to say, R is a binary relation – one uses the expressions

M ⊧ R(a, b)

and

M ⊧ aRb

interchangeably.

≅ |_{Mod(L)} denotes the isomorphism relation on elements in Mod(L). Here we understand isomorphism in the natural sense for classes of L-structures: two Lstructures M, N are isomorphic if there is some bijection π between these structures such for all relations R in L and a_{1}, ..., a_{k} ∈ M,

M ⊧ R(a_{1}, ..., a_{k})

if and only if

N ⊧ R(π(a_{1}), ..., π(a_{k})).

Note that Mod(L) is homeomorphic to ∏_{i∈ℕ} 2^{ℕ^{a(i)}}. Since each ℕ^{a(i)} (=_{df} {f|f : {1, 2, ..., a(i)} → ℕ}) is countable, we obtain that Mod(L) is homeomorphic to 2^{ℕ} and therefore Polish.

The definition is restricted to relational languages, but this entails no real loss of generality. If f is a function symbol appearing in the language of some model M we may pass to a new model M′ having a fresh relation R_{f} with the understanding

M ⊧ f(k_{1}, k_{2}, ...) = l ⇔ M′ ⊧ R_{f}(k_{1}, k_{2}, ..., l).

So for instance the axiom in group theory stating the existence of a right identity

∃e∀g(g · e = g)

would seamlessly reappear as

∃e∀g(R_{•}(g, e, g)).

Example 2.36. If L is generated by a single binary relation E, then Mod(L) can be thought of as the space of directed graphs having the natural numbers as the underlying set. Then M ∈ Mod(L) is the complete graph if

M ⊧ ∀a∀b(aEb).

Definition 2.37. The logic action of S_{∞} on Mod(L) for L a countable relational language is defined as follows. Given M ∈ Mod(L) and f ∈ S_{∞} we specify f · M by

f · M ⊧ R_{i}(k_{1}, k_{2}, ...) ⇔ M ⊧ R_{i}(f^{−1}(k_{1}), f^{−1}(k_{2}), ...).

In other words, we let (R_{i})^{f·M} be

{(f(k_{1}), f(k_{2}), ...) : (k_{1}, k_{2}, ...) ∈ (R_{i})^{M}}.

Note then that this action is continuous and ≅|_{Mod(L)} = E_{S_{∞}}^{Mod(L)}.

The manuscript below is a study of which orbit equivalence relations may in principle be drawn under the umbrella of the orbit equivalence relation induced by the logic action for some sufficiently rich language. This is not to say that it is a study of the logic action, or even of any sort of Polish group action. The object of investigation is one level of abstraction further removed, since in terms of developing a theory of invariants and classification we should be interested in the quotient of a space by an equivalence relation, not the details of how it arises.

Definition 2.38. An equivalence relation E is said to admit classification by countable models or admit classification by countable structures if for some countable language L

E ≤_{B}≅|_{Mod(L)};

in other words, we may in a Borel fashion assign to each element of the space X on which E is defined some L-structure considered up to isomorphism as a complete invariant.

In this context the terms model and structure are used more or less interchangeably.

Theorem 2.39. (Becker, Kechris) Let G be a closed subgroup of S_{∞} and X a Polish G-space. Then E_{G}^{X} admits classification by countable models.

Proof. [4] derives this from their result that for G = S_{∞} we can find a Borel S_{∞}-embedding from X to Mod(L). This proof is summarized at 6.19 and 7.18 below and another argument is implicit in §6.2.

Lemma 2.40. There is a countable language L with id(ℝ) ≤_{B}≅|_{Mod(L)}.

Proof. Let L be the language generated by {P_{q} : q ∈ ℚ}, each P_{q} is a unary predicate. Assign to r ∈ ℝ the L-structure M_{r} on ℕ such that for all n ∈ ℕ

(M_{r} ⊧ P_{q}(n)) ⇔ q < r.

Lemma 2.41. There is a countable language L with F_{2} ≤_{B}≅|_{Mod(L)}.

Proof. Let (P_{n})_{n∈ℕ} be a countable sequence of predicates. To each x ∈ 2^{ℕ×ℕ} assign the structure M_{x} on ℕ that has at each k, m, n ∈ ℕ

(M_{x} ⊧ P_{m}(3^{n}5^{k})) ⇔ x(n, m) = 1,

and

M_{x} ⊧ ¬P_{m}(l)

for l not of the form l = 3^{n}5^{k}. (Thus we associate to each strip x(n, ·) of x infinitely many a ∈ M_{x} such that at each m we have M_{x} ⊧ P_{m}(a) if and only if x(n, m) = 1.)

Any hereditarily countable set may be represented by a model for the language generated by a single binary relation R. So if A is hereditarily countable, and thus we can find an enumeration a⃗ = (a_{i})_{i∈ℕ} of

{A} ∪ TC(A) =_{df} {A} ∪ A ∪ ⋃ A ∪ ⋃ ⋃ A...,

where ⋃ A = {y_{1} : ∃y_{0} ∈ A(y_{1} ∈ y_{0})}, ⋃ ⋃ A = {y_{2} : ∃y_{1} ∈ ⋃ A(y_{2} ∈ y_{1})}, and so on, then we can let M(a⃗) be a model with

M(a⃗) ⊧ nRm

if and only if a_{n} ∈ a_{m}. The model M(a⃗) depends on the enumeration a⃗ = (a_{i})_{i∈ℕ}, since in general there is no reasonable method of choosing a⃗ purely from A; the isomorphism type of M(a⃗) however is a function of A alone.

We likewise obtain a parameterization of elements of HC (the hereditarily countable sets) by points in 2^{ℕ×ℕ}, since the structures for a single binary relation and with underlying set ℕ may be naturally identified with 2^{ℕ×ℕ}.

Exercise 2.42. Let L be generated by a single binary relation <. Let Mod(LO) be the set of M ∈ Mod(L) which are linear orderings – that is to say

∀a ∈ ℕ(M ⊧ ¬(a < a)),
∀a ≠ b ∈ ℕ(M ⊧ a < b ⇔ ¬(b < a)),
∀a, b, c ∈ ℕ(M ⊧ (a < b ∧ b < c) ⇒ a < c)).

Then Mod(LO) is a closed subset of Mod(L).

Exercise 2.43. Let L be a relational language. Let τ_{F O} be the topology on Mod(L) generated by taking as basic open all sets of the form

{M ∈ Mod(L) : M ⊧ φ(k_{1}, ..., k_{l})},

where φ is a formula from the language L and ⊧ is the satisfaction relation from 2.35. Show that (Mod(L), τ_{F O}) is a Polish space. (One may realize this as a G_{δ} subspace of Polish space. The main point is that the inductive definition of ⊧ for Mod(L) amounts to a G_{δ} set in a suitable topology. The hardest condition – namely M ⊧ ∃xφ(x) if and only if there is some k ∈ ℕ with M ⊧ φ(k) – is truly G_{δ}, while the others are only closed. See [26].)

Exercise 2.44. (i) Let L be a relational language and T a (countable set) of sentences in the language. Let B_{T} be the set of M ∈ Mod(L) such that for all φ ∈ T

M ⊧ φ.

Then B_{T} is a Borel set, and therefore a standard Borel space in the relative Borel structure. (A structure M ∈ B_{T} is said to be a model for T.)

(ii) Now let (φ_{n}(·))_{n∈ℕ} be a countable sequence of formulas in L. Then the set of M ∈ Mod(L) such that for all a ∈ ℕ there exists n with

M ⊧ φ_{n}(a)

is a Borel set.

(iii) Let L be generated by a single binary relation R. Let Mod(Gph) be the L-structures on ℕ which are (undirected) graphs; in other words, for M ∈ Mod(L) we have M ∈ Mod(Gph) if and only if

M ⊧ ∀x∀y(xRy ⇒ yRx).

Show that Mod(Gph) is closed in Mod(L), the connected graphs in Mod(Gph) are a G_{δ} subset of Mod(L), and the graphs with only finitely many components form a Borel subset.

2.4. Baire category methods

Definition 2.45. For X a Polish space we say that A ⊂ X is meager if it is disjoint from a dense G_{δ} and comeager if it includes one. Since every countable intersection of dense open sets is dense by the Baire category theorem we may choose to think of comeager sets as being large. In particular the class of comeager sets is closed under supersets and countable intersections. It is customary to write ∀^{∗}x ∈ X(“such and such about x”) if there is a comeager A ⊂ X on which we have that “such and such about x”; we write ∃^{∗}x ∈ X(“such and such about x”) if such and such on a non-meager set. A set B ⊂ X is said to have the Baire property if there is some O ⊂ X that is open and A ⊂ X comeager with A ∩ O = A ∩ B.

The sets with the Baire property form a σ-algebra, and so include the Borel.

Theorem 2.46. (Kuratowski, Ulam) Let X and Y be Polish. If B ⊂ X × Y is comeager (in the product topology) then for a comeager set of x ∈ X

B_{x} =_{df} {y : (x, y) ∈ B}

is comeager in Y; if B has the Baire property then B is non-meager if and only if there is a non-meager set of x ∈ X for which B_{x} is non-meager.

Proof. For the first proposition it suffices to consider the case that B is open dense. But if {U_{n} : n ∈ ℕ} is a basis for Y then for each n

O_{n} = {x ∈ X : U_{n} ∩ B_{x} ≠ ∅}

is open dense by assumption on B. Then B_{x} is comeager for all

x ∈ ⋂_{n∈ℕ} O_{n}.

The second part of the theorem follows from the first and 2.3: B non-meager implies there are open, non-empty U ⊂ X and V ⊂ Y with B ∩ U × V comeager in U × V. Since U and V are Polish B_{x} will be comeager in V for a comeager set of x ∈ U. Alternatively if B is meager in the product then C = (X × Y) \ B will be comeager, and we can conclude from the first part that C_{x} will be a comeager for a comeager collection of x ∈ X.

Thus if B ⊂ X × Y has the Baire property and

∀^{∗}x ∈ X∀^{∗}y ∈ Y ((x, y) ∈ B)

then

∀^{∗}x ∈ X(Y \ B_{x} must be meager),

implying B comeager in X × Y, and so

∀^{∗}y ∈ Y ∀^{∗}x ∈ X((x, y) ∈ B).

This is the right way to think about 2.46: Under reasonable assumptions the quantifiers ∀^{∗}x ∈ X and ∀^{∗}y ∈ Y commute.

Corollary 2.47. Let G be a Polish group, X a Polish G-space. If A ⊂ X is comeager, then ∀^{∗}x ∈ X∀^{∗}g ∈ G(g · x ∈ A).

Proof. Note that for all g ∈ G the set g^{−1} · A is comeager. We may assume A is Borel, in fact G_{δ}. Thus for

A = ⋂_{i∈ℕ} U_{i},

each U_{i} open and O_{i} the open set in G × X consisting of all (g, x) with g · x ∈ U_{i}, we have

{(g, x) ∈ G × X : g · x ∈ A} = ⋂_{i∈ℕ} O_{i}

to be Borel. From these observations and 2.46 we have that B = {(g, x) ∈ G × X : g · x ∈ A} is comeager in the plane. By a further application of 2.46 we see that ∀^{∗}x ∈ X(B_{x} is comeager).

Definition 2.48. Let X and Y be Polish spaces. A function f : X → Y is said to be Baire measurable if the preimage of every open set has the Baire property . The function f is said to be universally Baire measurable if given any Borel function g into X we have that f ∘ g is Baire measurable.

So in particular every Borel function is Baire measurable.

Lemma 2.49. Let f : X → Y be a Baire measurable function. Then there is a comeager set A ⊂ X on which f is continuous.

Proof. Since we choose A so f^{−1}(O) ∩ A is relatively open for all basic open O.

Lemma 2.50. Let G be a Polish group, X a Polish G-space, V ⊂ G open.

(i) If U ⊂ X is open then {x ∈ X : ∃^{∗}g ∈ V (g · x ∈ U)} is open. (ii) If A ⊂ X is G_{δ} then {x ∈ X : ∀^{∗}g ∈ V (g · x ∈ A)} is G_{δ}. (iii) If B ⊂ X is Borel then {x ∈ X : ∀^{∗}g ∈ V (g · x ∈ B)} is Borel.

Proof. For (i) we obtain from the continuity of the action that

{x ∈ X : ∃^{∗}g ∈ V (g · x ∈ U)}

equals

{x ∈ X : ∃g ∈ V (g · x ∈ U)}.

(ii) follows (i), since if we fix a countable basis B of G

∀^{∗}g ∈ V (g · x ∈ ⋂_{n∈ℕ} U_{n})

if and only if for all n ∈ ℕ and non-empty basic open W ⊂ V

∃^{∗}g ∈ W (g · x ∈ U_{n}).

For (iii): let C(X) be the collection of B ⊂ X for which

{x : ∀^{∗}g ∈ V (g · x ∈ B)}
{x : ∃^{∗}g ∈ V (g · x ∈ B)}

are both Borel whenever V ⊂ G open. We need to show C(X) forms a σ-algebra including the open sets.

First note that

Claim: If B ⊂ X is Borel and x ∈ X then

{g ∈ G : g · x ∈ B}

is Borel.

Proof of claim. Let D be the collection of subsets of X satisfying the conclusion of the claim. It includes the open sets and at once is closed under complementation. Since

{g ∈ G : g · x ∈ ⋂_{n∈ℕ} B_{n}}

= ⋂_{n∈ℕ}{g ∈ G : g · x ∈ B_{n}}

D forms a σ-algebra and thus includes all Borel sets. (Claim□)

Thus in particular for each x and Borel B ⊂ X and open V ⊂ G, the set of g ∈ V with

g · x ∈ B

has the Baire property , and thus

X \ {x : ∃^{∗}g ∈ V (g · x ∈ B)} = {x : ∀^{∗}g ∈ V (g · x ∈ (X \ B))}.

Thus C(X) is closed under complements. By (i) it includes the open sets and from the argument of (ii) it is closed under countable intersections.

The sets arising 2.50 appear so frequently that there is a further notation for describing them. This was introduced in [80].

Notation 2.51. Let X and G be as in 2.50 and V ⊂ G open. Then for any B ⊂ X we let B^{ΔV} = {x ∈ X : ∃^{∗}g ∈ V (g · x ∈ B)} and B^{∗V} = {x ∈ X : ∀^{∗}g ∈ V (g · x ∈ B)}. The sets B^{ΔV} and B^{∗V} are Vaught transforms of B.

Some authors also use B^{∗} for B^{∗G} and B^{Δ} for B^{ΔG}.

There are two essential facts about the Vaught transforms, both of which were implicit in 2.50: For B ⊂ X Borel,

X \ (B^{ΔV}) = (X \ B)^{∗V}

and if (B_{i})_{i∈ℕ} are Borel subsets of X and {V_{n} : n ∈ ℕ} an enumeration of the non-empty basic open sets in G, then

(⋃_{i∈ℕ} B_{i})^{ΔV} = ⋃_{i,j∈ℕ,V_{j}⊂V} B_{i}^{∗V_{j}}.

Definition 2.52. A subset A of a Polish space X is said to be Σ ∼_{1}^{1}, or analytic, if there is a Polish space Y and a Borel set B ⊂ Y and a Borel function f : Y → X with A = {f(y) : y ∈ B}. In other words, the Σ ∼_{1}^{1} sets are the Borel images of the Borel sets.

Theorem 2.53. (Luzin) Every Σ ∼_{1}^{1} set has the property of Baire. Every Σ ∼_{1}^{1} set is measurable with respect to any Borel measure.

Proof. We concentrate on the proof for the Baire property, with only a passing aside on measurability. For (A_{s})_{s∈ℕ^{<ℕ}} a collection of Borel sets in a Polish space X, indexed by finite functions of the form s: {0, 1, ..., n} → ℕ, let A((A_{s})_{s∈ℕ^{<ℕ}}) be the set of x ∈ X such that for some f ∈ ℕ^{ℕ}

x ∈ ⋂_{n∈ℕ} A_{f|_{{0,1,...,n}}}.

In other words, x ∈ A((A_{s})_{s∈ℕ^{<ℕ}}) if there is some sequence of choices s_{1} ∈ ℕ^{1}, s_{2} ∈ ℕ^{2},...s_{k} ∈ ℕ^{k},..., which are consistent in the sense of

s_{k} ⊂ s_{k+1}

all k and for which

x ∈ ⋂_{k∈ℕ} A_{s_{k}}.

Claim(I): For all B ⊂ X Borel there exists a family (A_{s})_{s∈ℕ^{<ℕ}} of closed sets with B = A((A_{s})_{s∈ℕ^{<ℕ}}).

Proof of Claim. Clearly closed sets can be represented in this form and the Borel sets may be obtained from the closed sets by iterating the operations of taking countable intersections and unions. Therefore it suffices to show that the class of sets that allow such representation are closed under countable unions and intersections.

So suppose (B_{i})_{i∈ℕ} are Borel sets and and at each n ∈ ℕ we have

A((A_{s}^{n})_{s∈ℕ^{<ℕ}}) = B_{n},

with each A_{s}^{n} closed, providing a representation of B_{n} as stated in the claim. If we let A_{n^{⌢}s} = A_{s}^{n} for each n ∈ ℕ and finite sequence s (where here n^{⌢}s is the finite sequence defined by n^{⌢}s(0) = n, n^{⌢}s(i + 1) = s(i)), then

⋃_{n∈ℕ} B_{n} = A((A_{s})_{s∈ℕ^{<ℕ}}).

For intersections we can partition ℕ into infinite subsets {C_{i} : i ∈ ℕ}, with each C_{i} enumerated in increasing order by (N(i, j))_{j∈ℕ}. Then for s: {0, 1, 2, ..., n} → ℕ with n = N(i, j) we may define A′_{s} to be A_{(s(N(i,0)),s(N(i,1)),...,s(N(i,j)))}^{i}; in other words, if the final element of domain of s is the jth element in C_{i} and we define

t : {0, 1, ..., j} → ℕ

by t(k) being equal to the the value s assumes on the kth element of C_{i}, then

A′_{s} = A_{t}^{i}.

We then have

⋂_{n∈ℕ} B_{n} = A((A′_{s})_{s∈ℕ^{<ℕ}}).

(Claim(I)□)

Claim(II): For all D ⊂ X with D ∈ Σ ∼_{1}^{1} there is a collection of closed sets (A_{s})_{s∈ℕ^{<ℕ}} in X with D = A((A_{s})_{s∈ℕ^{<ℕ}}).

Proof of claim: Let

f : Y → X

be a Borel function and

B ⊂ Y

a Borel set with f[B] = D. Note that

{(x, y) ∈ X × Y : y ∈ B, f(y) = x}

is Borel.

The idea of the proof of the claim is to produce (A_{s})_{s∈ℕ^{<ℕ}} such that an infinite function h ∈ ℕ^{ℕ} will encode three functions h_{0}, h_{1}, h_{2} ∈ ℕ^{ℕ}; these will be such that if x ∈ ⋂_{n} A_{h|_{n}}, then h_{2} will describe some y ∈ Y, h_{0} will witness (x′, y) ∈ B and f(y) = x′, for some x′ ∈ X, and h_{1} will certify that x′ = x.

By claim(I) there is a collection (C_{s})_{s∈ℕ^{<ℕ}} of closed sets with

{(x, y) ∈ X × Y : y ∈ B, f(y) = x} = A((C_{s})_{s∈ℕ^{<ℕ}}).

First let assume without loss of generality that C_{t} ⊂ C_{s} whenever s ⊂ t. Now for s ∈ ℕ^{n} define P r_{0}(s), P r_{1}(s), P r_{2}(s) ∈ ℕ^{[n/3]}, where [n/3] is the largest integer ≤ n/3, by

(P r_{0}(s))(k) = s(3k)
(P r_{1}(s))(k) = s(3k + 1)
(P r_{2}(s))(k) = s(3k + 2).

Let d_{0} be a complete metric on Y, d_{1} on X. For U ⊂ Y and V ⊂ X we let d_{0}(U) denote the supremum of {d_{0}(y_{0}, y_{1}) : y_{0}, y_{1} ∈ U} and d_{1}(V) denote the supremum of {d_{1}(x_{0}, x_{1}) : x_{0}, x_{1} ∈ V }. Choose (U_{s})_{s∈ℕ^{<ℕ}} open subsets of Y, (V_{s})_{s∈ℕ^{<ℕ}} open subsets of X, so that in both cases the empty function labels the whole space (i.e. U_{∅} = Y, V_{∅} = X), and for all n ∈ ℕ and s : {0, 1, ..., n − 1} → ℕ

d_{0}(U_{s}) < 2^{−n},

U_{s} = ⋃_{k∈ℕ} U_{s^{⌢}k},

d_{1}(V_{s}) < 2^{−n},

V_{s} = ⋃_{k∈ℕ} V_{s^{⌢}k},

(where here for s ∈ ℕ^{n} we define s^{⌢}k ∈ ℕ^{n+1} by s^{⌢}k|_{n} = s but s^{⌢}k(n) = k). For n ∈ ℕ and s : {0, 1, ..., n − 1} → ℕ let A_{s} be the set of x ∈ X for which

(i) x ∈ \overline{V_{P r_{1}(s)}} (where \overline{V_{P r_{1}(s)}} indicates the closure of V_{P r_{1}(s)}); (ii) there exists x′, y′ with

(iia) (x′, y′) ∈ C_{P r_{0}(s)}; (iib) x′ ∈ \overline{V_{P r_{1}(s)}}; (iic) y′ ∈ \overline{U_{P r_{2}(s)}}.

Thus if x ∈ A((A_{s})_{s∈ℕ^{<ℕ}}) we may find
x_{i} → x,

w ∈ ℕ^{ℕ}, and a Cauchy sequence (y_{i})_{i∈ℕ} in Y with

(x_{i}, y_{i}) ∈ C_{w|_{{0,1,...,j}}} =_{df} C_{w|_{j+1}}

for all j < i; so if y_{∞} is the limit of (y_{i})_{i∈ℕ} then for all j

(x, y_{∞}) ∈ C_{w|_{{0,1,...,j}}}.

Hence y_{∞} ∈ B and x = f(y_{∞}) will be in D.

Conversely if x ∈ D and x = f(y), then by the assumption on the (C_{s})_{s∈ℕ^{<ℕ}} we can first choose w ∈ ℕ^{ℕ} such that

(x, y) ∈ ⋂_{n∈ℕ} C_{w|_{n}}.

Then a direct appeal to the definitions enables us to choose h ∈ ℕ^{ℕ} with

x ∈ ⋂_{n∈ℕ} A_{h|_{n}}.

(Claim(II)□)

Returning to the theorem, fix closed sets A_{s} for s ∈ ℕ^{<ℕ} such that

A = A((A_{s})_{s∈ℕ^{<ℕ}})

and attempt to show A has the Baire property . Assume that A_{s} ⊃ A_{t} for s ⊂ t. For finite sequences s and t let s^{⌢}t be the sequence obtained by following s by t – so that s^{⌢}t(i) = s(i) for i less than the length of i, and s^{⌢}t(l + i) = t(i) for l equal to the length of s. Note then that A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) is in some sense equal to the points we can obtain in A given that we have decided to work below s.

Fix a countable basis for X. Let O_{s} equal the union of all the basic open sets U such that A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) ∩ U meager, and let C_{s} = X \ O_{s}. (For the measure version of this theorem, we produce Borel C_{s} ⊃ A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) with the same outer measure as A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) and C_{s} ⊂ A_{s}. This in particular implies that any measurable B ⊂ C_{s} disjoint from A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) is null.)

Since each A_{s} is closed, we have C_{s} ⊂ A_{s} at each s ∈ ℕ^{<ℕ}.

Let

M_{0} = ⋃_{n∈ℕ} ⋃_{s∈ℕ^{n}} C_{s} \ (⋃{C_{t} : t ∈ ℕ^{n+1}, t ⊃ s}).

Claim(III): M_{0} is meager.

Proof of claim: Suppose instead that some

C_{s} \ (⋃{C_{t} : t ∈ ℕ^{n+1}, t ⊃ s})

is relatively comeager in some basic non-empty open U ⊂ X. Then at each k ∈ ℕ we have from the definition of C_{s^{⌢}k} that

U ∩ A((A_{s^{⌢}k^{⌢}t})_{t∈ℕ^{<ℕ}})

is meager in U, and hence

A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) = ⋃_{k∈ℕ} A((A_{s^{⌢}k^{⌢}t})_{t∈ℕ^{<ℕ}})

is meager in U, and thus U∩C_{s} is empty, as required for a contradiction. (Claim(III)□)

Let

M_{1} = ⋃_{s∈ℕ^{<ℕ}} A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) \ C_{s}.

Claim(IV): M_{1} is meager.

Proof of claim: From the definition of C_{s} we have that it is a countable union of meager sets of the form A((A_{s^{⌢}t})_{t∈ℕ^{<ℕ}}) ∩ U, and hence meager. (Claim(IV)□)

Claim(V): C_{∅}ΔA =_{df} (A \ C_{∅}) ∪ (C_{∅} \ A) is a subset of M_{0} ∪ M_{1}.

Proof of claim: If x ∈ C_{∅} and x ∉ M_{0}, then invoking the definition of M_{0} allows us to successively choose increasing finite sequences s_{0} ⊂ s_{1}.... ⊂ s_{n} ⊂ s_{n+1}... such that at each n

x ∈ C_{s_{n}}
∴ x ∈ A_{s_{n}}.

Conversely, if x ∈ A = A((A_{s})_{s∈ℕ^{<ℕ}}) but not in M_{1} we have x ∈ C_{∅} by the definition of M_{1}. (Claim(V)□)

Claim(V) implies that since C_{∅} has the the Baire property so too does A.

Theorem 2.54. (Jankov, von Neumann) Let X, Y be Polish and A ⊂ X × Y in Σ ∼_{1}^{1}. Then A can be uniformized by a function that is both Baire and universally measurable, in the sense that for some h : X → Y we have that for all x and y

(x, y) ∈ A ⇒ (x, h(x)) ∈ A

with h^{−1}[U] having the Baire property and being measurable with respect to any Borel measure for all open U ⊂ X.

Proof. The idea of the proof is to define for each x a canonical h(x) and f(x) such that

(x, h(x)) ∈ ⋂_{n∈ℕ} A_{f(x)|_{n}}.

This will be defined using what is in some sense the “least” possible choice for h(x) and each (f(x))(n).

Suppose A = A((A_{s})_{s∈ℕ^{<ℕ}}), each A_{s} closed. Let {U_{l} : l ∈ ℕ} be a basis for Y and d a complete metric. For x ∈ X in the projection of A define N_{x,n}, M_{x,n} ∈ ℕ so that at each n ∈ ℕ we choose N_{x,n} least so that there exists f ∈ ℕ^{ℕ} and

y ∈ ⋂_{m<n} U_{M_{x,m}}

with

(x, y) ∈ ⋂_{k∈ℕ} A_{f|_{k}}

(where here f|_{k} stands for f|_{{0,1,...,k−1}}, the restriction of f to its first k values) and such that for all i ≤ n

f(i) = N_{x,i};

if there is no y with (x, y) ∈ A just let N_{x,n} and M_{x,n} always assume the default value 0. Then let M_{x,n} be least such that d(U_{M_{x,n}}) < 2^{−n}, and \overline{U_{M_{x},n}} ⊂ U_{M_{x},n−1}, and there is some f ⊃ (N_{x,0}, N_{x,1}, ...N_{x,n}) (that is to say f extends the finite sequence whose first term is N_{x,0}, whose second term is N_{x,1}, ..., and so on, with final term N_{x,n}) and there is some

y ∈ ⋂_{m≤n} U_{M_{x,m}}

with

(x, y) ∈ ⋂_{k∈ℕ} A_{f|_{k}}.

By induction on n we see that for each s, t ∈ ℕ^{{0,1,...,n−1}} the set {x : ∀i < n(M_{x,i} = s(i), N_{x,i} = t(i))} is in the σ-algebra generated by the Σ ∼_{1}^{1} sets. Thus if we let h(x) ∈ ℕ^{ℕ} be given by

{h(x)} = ⋂_{m∈ℕ} U_{M_{x,m}}

then we obtain that for any open U the pullback h^{−1}[U] is again in the least σ-algebra containing the Σ ∼_{1}^{1} sets and therefore is measurable and has the Baire property .

For any x for which there exists y with (x, y) ∈ A we obtain by induction on n that at each stage

∃y_{n} ∈ ⋂_{m<n} U_{M_{x,m}}

with

(x, y_{n}) ∈ ⋂_{k≤n+1} A_{(N_{x,0},N_{x,1},...N_{x,k})}.

Then since

y_{n} → h(x)

and each A_{s} is closed we obtain

(x, h(x)) ∈ ⋂_{k∈ℕ} A_{(N_{x,0},N_{x,1},...,N_{x,k})}.

Theorem 2.55. (Novikov) Let A ⊂ X × Y be Σ ∼_{1}^{1}, X and Y Polish. Then the sets {x ∈ X : ∀^{∗}y ∈ Y ((x, y) ∈ A)} and {x : ∃^{∗}y ∈ Y ((x, y) ∈ A)} are Σ ∼_{1}^{1}.

Proof. Recalling claim(II) from the proof of 2.53, suppose A = A((A_{s})_{s∈ℕ^{<ℕ}}), each A_{s} closed, A_{t} ⊂ A_{s} whenever t ⊃ s. Assume A_{∅} = X × Y. Following 2.54 and 2.49, for x ∈ X,

∀^{∗}y ∈ Y ((x, y) ∈ A)

if and only if there is a comeager C ⊂ Y and continuous f : C → ℕ^{ℕ} so that for all y ∈ C and n ∈ ℕ

(x, y) ∈ A_{f(y)|_{n}}.

With further manipulation of the definitions we show the set of x for which such C and f exist to be Σ ∼_{1}^{1}:

Fix enumerations {U_{n} : n ∈ ℕ} and {W_{n} : n ∈ ℕ} of the non-empty basic open sets in Y and X. Let d be a complete metric on Y. Let B be the set of all (x, h) ∈ X × (ℕ^{<ℕ})^{ℕ} such that

(i) h(l) = s implies that for all W_{n} containing x there exists

y′ ∈ U_{l}

and

x′ ∈ W_{n}

with

(x′, y′) ∈ A_{s};

(ii) for all U_{k} nonempty and m ∈ ℕ there exists some l ∈ ℕ, n > m, and s : {0, 1, ..., n} → ℕ with

d(U_{l}) < 2^{−m},
U_{l} ⊂ U_{k}

and

h(l) = s;

(iii) U_{l} ⊂ U_{k} and h(l) = s, h(k) = t imply s ⊃ t. This set B is Borel; in fact, (ii) is no worse than requiring (x, h) be in a G_{δ} set while (i) and (iii) are closed.

Claim: ∃h((x, h) ∈ B) if and only if ∀^{∗}y ∈ Y ((x, y) ∈ A).

Proof of claim: First suppose (x, h) ∈ B. For each m let O_{m} be the union of open sets U_{l} for which

d(U_{l}) < 2^{−m}

and there exists s: {0, 1, ..., n} → ℕ for some n > m with

h(l) = s.

Let

C_{h} = ⋂_{m∈ℕ} O_{m}.

Condition (ii) indicates that this set must be comeager. In virtue of (iii) we have that h defines a continuous f_{h} : C_{h} → ℕ^{ℕ} by

f_{h}(y) ⊃ s ⇔ ∃l(y ∈ U_{l} ∧ h(l) ⊃ s).

With (i) we have that whenever y ∈ C_{h} we obtain y_{i} → y and x_{i} → x with each

(x_{n}, y_{n}) ∈ A_{f_{h}(y)|_{n}}

and hence

⋂_{n∈ℕ}(x, y) ∈ A_{f_{h}(y)|_{n}}.

Conversely if ∀^{∗}y ∈ Y ((x, y) ∈ A) then we may by 2.54 and 2.49 find a comeager C ⊂ Y and continuous f : C → ℕ^{ℕ} such that for all y ∈ C

(x, y) ∈ ⋂_{n∈ℕ} A_{f_{h}(y)|_{n}}.

Then we define h by the specification that h(l) = s if s ∈ ℕ^{≤l} is as long as possible subject to the requirement that f(y) ⊃ s all y ∈ U_{l} ∩ C. (□Claim)

Thus for π_{0} : X × (ℕ^{<ℕ})^{ℕ} → X given by (x, h) ↦ x

{x ∈ X : ∀^{∗}y ∈ Y ((x, y) ∈ A)} = π_{0}[B].

The case of ∃^{∗} is similar.

Lemma 2.56. (Pettis) Let G be a Polish group.

(i) If A ⊂ G with the property of Baire and non-meager then AA^{−1} contains an open neighborhood of the identity.

(ii) Let H be a Polish group, π : G → H be a Borel homomorphism. Then π is continuous.

(iii) Moreover if π is onto then it is open.

(iv) If A, B ⊂ G are both non-meager with the Baire property then AB contains an open set.

Proof. (i): Choose U an open neighborhood of the identity and g ∈ G with A comeager in Ug, and then W, V open neighborhoods of the identity with

W^{−1}V ⊂ U,

which amounts to asserting that for all ϵ ∈ W

V ⊂ ϵU
∴ V g ⊂ ϵUg.

Thus for all ϵ ∈ W the translation ϵA is still comeager in V g. Thus there exists some

δ ∈ ϵA ∩ A

and

∴ ϵ ∈ δA^{−1} ⊂ AA^{−1}.

(ii) is a corollary of (i): If U is any non-empty set in H and Q ⊂ H is a countable dense subset of π[G] then

Q · U ⊃ π[G].

ϵ  Ug ϵUg  U  g V  ϵ^{−1}V V g

Figure 2.1. The shifts of U

Hence for Q′ ⊂ G countable with π[Q′] = Q we have

Q′ · π^{−1}[U] ⊃ G

and thus for some g ∈ G the set g · π^{−1}[U] is nonmeager, and hence π^{−1}[U] is nonmeager.

For every W open neighborhood of the identity in H we can find U open containing the identity with U^{−1}U ⊂ W, and then since π^{−1}[U] nonmeager, π^{−1}[W] contains an open set. More generally given an arbitrary open set W ⊂ H and g ∈ π^{−1}[W], we can find open neighborhood O of 1 in G such that

O ⊂ π^{−1}[π(g^{−1})W] = g^{−1}π^{−1}[W]
∴ gO ⊂ π^{−1}[W].

(iii) resembles (ii).

(iv): We can find open neighborhood V of the identity and g, h ∈ G for which gA is relatively comeager in V and Bh is relatively comeager in V. It suffices to show then that gABh includes an open set, which in turn follows by the proof of (i).

So in particular a Borel group has at most one Polish group topology compatible with its Borel and algebraic structure.

Definition 2.57. A subset P of a Polish space is perfect if it is closed, nonempty, and has no isolated points.

For instance, the unit interval is perfect; the Cantor space 2^{ℕ} is perfect.

Theorem 2.58. (Kuratowski; Mycielski) Let X be a Polish space without isolated points and suppose that for each n ∈ ℕ the relation

R_{n} ⊂ X^{n}

is meager as a subset of the n-fold product of X. Then there is a perfect P ⊂ X such that for all distinct x_{1}, ..., x_{n} ∉ P

(x_{1}, ..., x_{n}) ∉ R_{n},

and continuous bijection π : 2^{ℕ} → P.

Proof. Just for the case of a single R ⊂ X × X; the general case is similar. Choose a sequence of open dense sets {O_{n} : n ∈ ℕ} in X × X such that

R ∩ ⋂_{n∈ℕ} O_{n} = ∅

O_{n+1} ⊂ O_{n}.

Let d be a complete metric on X; in what follows recall the notation that s ∈ 2^{n} indicates that s : {0, 1, ..., n − 1} → {0, 1} is a function from the first n elements of ℕ to {0, 1}. We now choose open {U_{s} : s ∈ 2^{<ℕ}} in X level by level as follows:

(i) U_{s} nonempty for all s;

(ii) for s ⊂ t

U_{t} ⊂ U_{s};

if moreover s ≠ t then \overline{U_{t}} (the closure of U_{t}) is included in U_{s}; (iii) for s ≠ t ∈ 2^{n}

U_{s} × U_{t} ⊂ O_{n};

(iv) for s ∈ 2^{n}

d(U_{s}) < 2^{−n},

where d(U_{s}) is sup{d(x, y) : x, y ∈ U_{s}};

(v) U_{s} ∩ U_{t} nonempty implies s ⊂ t or t ⊂ s. Once all this has been done just take

P = ⋃_{f∈2^{ℕ}} ⋂_{n∈ℕ} U_{f|_{n}}.

For any f ∈ 2^{ℕ} we can define π(f) with

{π(f)} = ⋂_{n∈ℕ} U_{f|_{n}} ≠ ∅.

Thus π : 2^{ℕ} → P is one to one, continuous, and onto, so P must be perfect. By (iii) f ≠ f′ implies

(π(f), π(f′)) ∈ ⋂_{n∈ℕ} O_{n}

∴ (π(f), π(f′)) ∉ R. As for showing that this construction can be completed, suppose inductively we

have built (U_{s})_{s∈2^{n}} for some n ∈ ℕ. Choose V_{s}^{0}, V_{s}^{1} ⊂ U_{s} open disjoint nonempty subsets with diameter less than 2^{−n−1} and with closures included in U_{s} – and this is possible by the assumption that X has no isolated points. Let (t(i), s(i))_{i<K(n)} be an enumeration of all pairs of distinct functions from {0, 1, ..., n} to {0, 1}. We build W_{t}^{i} by induction on i ≤ K(n), starting with

W_{t}^{0} = V_{s}^{0} if t = s^{⌢}0 for some s and otherwise

W_{t}^{0} = V_{s}^{1} if t = s^{⌢}1 for some s. Given (W_{s}^{i})_{s∈2^{n+1}} one finds W_{s(i)}^{i+1} ⊂ W_{s(i)}^{i} and W_{t(i)}^{i+1} ⊂ W_{t(i)}^{i} so that

W_{s(i)}^{i+1} × W_{t(i)}^{i+1} ⊂ O_{n}.

This is possible by O_{n} being dense open. For t ≠ s(i), t(i) just set W_{t}^{i+1} = W_{t}^{i}. Finally with (U_{t})_{t∈2^{n+1}} = (W_{t}^{K(n)})_{t∈2^{n+1}} we continue the process one further step.

Corollary 2.59. Let E be a meager equivalence relation on a Polish space X. Then

id(2^{ℕ}) ⊑_{c} E.

Proof. Since id(X) must therefore be meager in X × X, the Polish space can have no isolated points and so we can apply 2.58.

Exercise 2.60. Let f : X → Y be Borel and A ⊂ Y a Σ ∼_{1}^{1} set. Then f^{−1}[A] ∈ Σ ∼_{1}^{1}.

Exercise 2.61. (i) The collection of Σ ∼_{1}^{1} subsets of a Polish space are closed under countable intersections and unions.

(ii) Let (A_{n})_{n∈ℕ} be a sequence of Borel subsets of a product of Polish spaces X × Y. Then the set of x ∈ X such that

∀n ∈ ℕ ∃y ∈ Y ((x, y) ∈ A_{n})

is Σ ∼_{1}^{1}.

(iii) If A ⊂ X_{1} × X_{2} × .... × X_{n} × X_{n+1} is a Σ ∼_{1}^{1} subset of a product of Polish spaces, then

{x⃗ ∈ X_{1} × ... × X_{n} : ∃a ∈ X_{n+1}(x^{⌢}a ∈ A)}

is again Σ ∼_{1}^{1}.

Exercise 2.62. If G is a Polish group and H < G a Borel subgroup, then H has countable index (i.e. |G/H| ≤ ℵ_{0}) if and only if H is clopen.

Exercise 2.63. There is a perfect P ⊂ ℝ such that all distinct x_{1}, x_{2}, ..., x_{n} ∈ P are algebraically independent over ℚ.

Exercise 2.64. Show that if F is an equivalence relation on a Polish space with every equivalence class countable, then

F_{2} ≰_{B} F.

(By analogy with 2.31, for θ : B_{2} → X a Borel function it suffices to use 2.46, 2.47, and 2.49 to find some comeager C ⊂ B_{2} on which the function θ|_{C} is continuous, and then to find some x ∈ B_{2} so that ∀^{∗}g ∈ S_{∞} and for all finite F ⊂ ℕ the set

C ∩ {y ∈ 2^{ℕ×ℕ} : ∀i ∈ F (y(i, ·) = g · x(i, ·))}

is relatively comeager in

{y ∈ 2^{ℕ×ℕ} : ∀i ∈ F (y(i, ·) = g · x(i, ·))},

in the sense that if we view 2^{ℕ×ℕ} as decomposed into 2^{F ×ℕ} × 2^{(ℕ\F)×ℕ} then

∀^{∗}z ∈ 2^{(ℕ\F)×ℕ}(g · x|_{2^{F ×ℕ}}, z) ∈ C.

This method of approximating Borel reductions by continuous reappears at 3.14 and 3.18.)

Exercise 2.65. Let E ⊂ X × X be a Σ ∼_{1}^{1} equivalence relation.

(i) E is meager if and only if every equivalence class is meager.

(ii) If E has only countably many classes then for all U ⊂ X open and nonempty there is some V ⊂ U open and non-empty with E comeager in V × V.

There is a non-trivial notion of the potential Borel complexity of an orbit equivalence relation. For the case of G a closed subgroup of S_{∞} this was exactly the subject of [41].

Exercise 2.66. Let G be a Polish group and X a Polish G-space.

(i) Suppose F is a G_{δσ}-equivalence relation and θ is a Borel function witnessing E_{G}^{X} ≤_{B} F. Then for C ⊂ X a dense G_{δ} on which θ is continuous and x ∈ X such that ∀^{∗}g ∈ G(g · x ∈ C) we have [x]_{G} ∈ G_{δσ}.

(ii) Show that c_{0} is not G_{δσ} by observing that if A ⊂ c_{0} is G_{δ} in the product topology on ℝ^{ℕ} then A is nowhere dense in the sup norm topology on c_{0}, and hence meager as a subset of c_{0}. (If A = ⋂_{i∈ℕ} U_{i} is a G_{δ} subset of c_{0} not nowhere dense in the topology of c_{0}, then after possibly translating, we may assuming A is dense inside

{x⃗ ∈ c_{0} : sup|x_{i}| < ϵ},

for some ϵ > 0. Then we may continually choose k_{0} < k_{1} < ... < k_{n} < k_{n+1}... and bounded open sets V_{n} ⊂ [−ϵ, ϵ]^{k_{n}} such that for any

(z_{0}, z_{1}, ...) = z⃗ ∈ \overline{V_{n+1}}

we have z⃗|_{k_{n}} ∈ V_{n},

|z_{k_{n}}| ≥ \frac{ϵ}{2},

and for any x⃗ ∈ ℝ^{ℕ} with x⃗|_{k_{n}} ∈ V_{n} we have

x⃗ ∈ U_{n}.

Now compactness allows us to choose y⃗ so that at every n

y⃗|_{k_{n}} ∈ \overline{V_{n}}
∴ y⃗|_{k_{n−1}} ∈ V_{n−1}.

y⃗ cannot be in c_{0} since at each n we have |y_{k_{n}}| ≥ ϵ/2, which contradicts y⃗ ∈ A = ⋂_{i∈ℕ} U_{i}.)

(iii) Let us say that an orbit equivalence relation is potentially G_{δσ} if it is ≤_{B}- reducible to some G_{δσ} equivalence relation. From (ii) and (iii) we have that the coset equivalence relation of c_{0} in ℝ^{ℕ} is not potentially G_{δσ}.

36 2. DEFINITIONS AND TECHNICALITIES

CHAPTER 3 Turbulence

Here we discuss various sufficient conditions for non-classifiability. §3.1 begins with smoothness, proving that properly generically ergodic group actions induce non-smooth orbit equivalence relations. This is the model for §3.2, which introduces the more elaborate notion of turbulence and in turn establishes radical non-classifiability results for actions in this class. §3.3 presents some examples and applications. §3.4 consists in historical remarks.

3.1. Generic ergodicity

Definition 3.1. Let G be Polish group and X a Polish G-space. The action is said to be properly generically ergodic if:

(i) every orbit is dense; and (ii) every orbit is meager.

By analogy we may say that the orbit equivalence relation is properly generically ergodic if it arises, or can be made to arise, from such an action.

Theorem 3.2. (classical) Suppose G is a Polish group, X is a Polish G-space, and every orbit in X is dense. Suppose θ : X → ℝ is Baire measurable and Ginvariant (in the sense that ∀x ∈ Xg ∈ G(θ(x) = θ(g · x))). Then there is a comeager set C ⊂ X on which θ is constant.

Proof. Let C be a comeager set on which θ is continuous. Let x ∈ X be such that ∀^{∗}g ∈ G(g · x ∈ C); x exists by 2.47. Then [x]_{G} ∩ C is dense in C, so by continuity θ is constant on C.

Thus we obtain a sufficient condition for an orbit equivalence relation to be non-smooth.

Corollary 3.3. Suppose G is a Polish group and X is a properly generically ergodic Polish G-space. Then X/G ≰_{B} id(ℝ).

In the proof of 3.2 we did not need every orbit dense; only a comeager set of dense orbits. But if one orbit is dense then generically so are all.

Lemma 3.4. Suppose G is a Polish group and X is a Polish G-space. Suppose some orbit is dense. Then the set of dense orbit is an invariant dense G_{δ}.

Proof. Let {U_{n} : n ∈ ℕ} enumerate the non-empty basic open sets in X. Then for each l the set of x ∈ X such that

[x]_{G} ∩ U_{l} ≠ ∅

is invariant and open. Thus the set of dense orbits is an invariant G_{δ}.

Therefore as soon as we know this set to be non-empty we are done.

The next statement foreshadows lemmas from §3.2.

Corollary 3.5. Suppose G is a Polish group and X is a Polish G-space. Suppose some orbit is dense and every orbit is meager.

Then there is an invariant dense G_{δ} set X_{0} ⊂ X such that X_{0} is a properly generically ergodic Polish G-space. In particular E_{G}^{X} ≰_{B} id(ℝ).

Proof. Let X_{0} be the set of points with dense orbits, and note the notions of categoricity coincide between X and X_{0} since X_{0} is comeager by 3.4.

By the same proof we obtain E_{G}^{X} ≰_{B} id(Y) for any Polish space Y.

Definition 3.6. Let E and F be equivalence relations on X and Y. E is said to be generically F -ergodic if whenever θ : X → Y is Baire measurable with ∀x_{1}, x_{2} ∈ X,

x_{1}Ex_{2} ⇒ θ(x_{1})F θ(x_{2}),

then we must have some comeager set C ⊂ X such that for all x_{1}, x_{2} ∈ C

θ(x_{1})F θ(x_{2}).

So 3.2 states that any properly generically ergodic orbit equivalence relation is id(ℝ)-ergodic, and in particular, since every orbit is meager, the equivalence relation is non-smooth.

Exercise 3.7. Let G be a countable infinite group. Show that 2^{G}/G (in the shift action from 2.24) is not smooth. (Hint: The main issue is the existence of some dense orbit. But given finite F, F′ ⊂ G and basic open sets

U = {f ∈ 2^{G} : ∀g ∈ F (f(g) = i_{g})},
U′ = {f ∈ 2^{G} : ∀g ∈ F′(f(g) = j_{g})},

where each i_{g}, j_{g} ∈ {0, 1}, we may obtain h ∈ G such that h · F ∩ F′ = ∅ and hence h · U ∩ U′ ≠ ∅. This point granted, a diagonalization argument enables us to produce a sequence of non-empty basic open sets (U_{i})_{i∈ℕ}, each

\overline{U_{i+1}} ⊂ U_{i},

and

d(U_{i}) < 2^{−i},

with respect to a complete metric d on 2^{G}, so that for any basic open set W we have some i and h ∈ G with h · U_{i} ⊂ W. For {f} = ⋂ U_{i} we have [f]_{G} dense.)

Exercise 3.8. Show: id(ℝ) <_{B} E_{0}; id(ℝ) <_{B} ℝ/ℚ; id(ℝ) <_{B} 2^{ℤ}/ℤ. (3.3 is applicable to E_{0}, since this equivalence relation arises from an action of the infinite rank ℤ_{2}-module on 2^{ℕ}; compare for instance 7.9.)

We will see in §7 that if X is a properly generically ergodic Polish G-space then E_{0} is continuously reducible to E_{G}^{X}.

Exercise 3.9. Let G be a Polish group and let X be a Polish G-space. Show that if E_{G}^{X} is generically id(ℝ)-ergodic, then

∀^{∗}x ∈ X([x]_{G} is dense).

Exercise 3.10. The isomorphism relation on rank one torsion free abelian groups is not smooth. Before proving or even precisely stating this it is necessary to work out some definitions:

(i) (The set of models displaying group structures on ℕ forms a Polish S_{∞}- space.) Let R_{•}(·, ·, ·) be a three placed relation and let R_{−1}(·, ·) be a two placed relation and let L be the language generated by these two. Let Mod(Grp) ⊂Mod(L) be the set of M for which:

(a) for all a, b ∈ ℕ there exists c, d ∈ ℕ so that

M ⊧ R_{•}(a, b, c) ∧ R_{−1}(a, d)

(b) for all a, b, c, c′, d, d′ ∈ ℕ

M ⊧ (R_{•}(a, b, c) ∧ R_{•}(a, b, c′)) ⇒ c = c′
M ⊧ (R_{−1}(a, d) ∧ R_{−1}(a, d′)) ⇒ d = d′;

and if we therefore define (•(·, ·))^{M} : ℕ×ℕ → ℕ and ((·)^{−1})^{M} : ℕ → ℕ accordingly, so that for all a, b

M ⊧ R_{•}(a, b, (•(a, b))^{M})
M ⊧ R_{−1}(a, ((a)^{−1})^{M})

then (c) (•(·, ·))^{M} defines a group operation on ℕ with inverse ((·)^{−1})^{M} (so that for all a, b, c, d, e, f, g associativity amounts to

M ⊧ (R_{•}(a, b, d) ∧ R_{•}(d, c, e) ∧ R_{•}(b, c, f) ∧ R_{•}(a, f, g)) ⇒ g = e

while the defining condition on the inverse for a group comes out as for all a, b, c, d

M ⊧ (R_{−1}(a, b) ∧ R_{•}(a, b, c)) ⇒ (R_{•}(c, d, d) ∧ R_{•}(d, c, d)),
M ⊧ (R_{−1}(a, d) ⇒ R_{−1}(d, a));

Show that Mod(Grp) is an invariant G_{δ} subset of Mod(L) and thereby forms a Polish S_{∞}-space.

(ii) (The torsion free abelian rank 1 groups form a Polish S_{∞}-space.) Let TFA_{1} consist of all M ∈ Mod(Grp) such that:

(d) (abelian) for all a, b, d, d′ ∈ ℕ

M ⊧ (R_{•}(a, b, d) ∧ R_{•}(b, a, d′)) ⇒ d = d′;

(e) (torsion free) for all a = a_{0}, a_{1}, ...a_{n} ∈ M such that at each i < n

M ⊧ R_{•}(a, a_{i}, a_{i+1})
M ⊧ R_{•}(a_{n}, a_{n}, a_{n})

we have

M ⊧ R_{•}(a, a, a);

(f) (rank one) for all a, b ∈ ℕ there are a_{1} = a, a_{2}, ..., a_{n} and b_{1} = b, b_{2}, ..., b_{k} so that at each i < n

M ⊧ R_{•}(a, a_{i}, a_{i+1})

and for i < k

M ⊧ R_{•}(b, b_{i}, b_{i+1})

and

a_{n} = b_{k}

(in other words n · a = k · b). Show that TFA_{1} is an invariant G_{δ} subset of Mod(Grp).

(iii) Let (ℚ^{+}, ·) be the multiplicative group of the positive rationals. Let 2^{ℚ} be the set of f : ℚ → {0, 1} with the usual product topology, so that basic open sets have the form

{f ∈ 2^{ℚ} : f(q_{1}) = i_{1}, ..., f(q_{n}) = i_{n}}.

Let X be the space of f ∈ 2^{ℚ} for which the set A_{f} of q ∈ ℚ with f(q) = 1 forms a subgroup of (ℚ, +). Let (ℚ^{+}, ·) act on X by

q̂ · f(q) = f(qq̂).

(iv) Show that X is a Polish (ℚ^{+}, ·)-space.

(v) Use 3.5 to show that E_{(ℚ^{+},·)}^{X} is non-smooth.

(vi) Show that X/(ℚ^{+}, ·) ≤_{B} TFA_{1}/S_{∞} and hence conclude that isomorphism relation on elements of Mod(L) corresponding to torsion free rank 1 abelian groups is non-smooth.

(vii) On the other hand, show that the action of S_{∞} on TFA_{1} is not properly generically ergodic since there is a dense G_{δ} orbit.

It is implicit in [1] that we conversely obtain TFA_{1}/S_{∞} ≤_{B} E_{0}. Since E_{0} can be thought of as a kind of Borel instantiation of the subsets of ℕ considered up to finite difference, this is tantamount to classifying rank one torsion free abelian groups by elements of P(ℕ)/Finite. The method of proof for this is to take a rank one torsion free abelian group G with underlying set ℕ and assign k_{G} ∈ ℕ to be the first non-zero element of G. One then defines a function θ(G) in 2^{ℕ×ℕ}, where for p_{n} the nth prime we let (θ(G))(m, n) = 1 if and only if (p_{n})^{m} divides k_{G}. With some effort one can show that for any two G_{1} and G_{2} we have θ(G_{1})E_{0}(2^{ℕ×ℕ})θ(G_{2}) if and only if G_{1} ≅ G_{2}.

Exercise 3.11. The action of S_{∞} on itself by conjugation is smooth. (The number of n-cycles for n ∈ {1, 2, 3, ...., ∞} is a complete invariant.)

The next exercise may seem artificial. The point is that this is exactly kind of reduction one obtains by the Ulm invariants (see [48]) for countable abelian pgroups. The definition itself is technical, and does not play an important part until chapter 9.

Exercise 3.12. Recall that ω_{1} is the first uncountable ordinal, so that (ω_{1}, <) is an uncountable well ordered set all of whose proper initial segments are countable. Let 2^{<ω_{1}} be the set of all bounded subsets of ω_{1} – that is to say the A ⊂ ω_{1} for which there is some α ∈ ω_{1} such that for all β ∈ A we have β < α.

Let us say that a function θ : X → 2^{<ω_{1}} from a standard Borel space space to the bounded subsets of ω_{1} is UBMC (short for universally Baire measurable in the codes) if we have the following:

whenever g : Y → X is a Borel function from a Polish space there will be some α ∈ ω_{1} and comeager C ⊂ Y such that:

(i) for all y ∈ C

θ(g(y)) ⊂ {β ∈ ω_{1} : β < α};

(ii) if π : ℕ → {β ∈ ω_{1} : β < α} is a bijection, then the function

C → 2^{ℕ}
y ↦ {n ∈ ℕ : π(n) ∈ θ(g(y))}

is Borel. (The issue of this function being Borel does not depend on the choice of π.)

Let us say that an equivalence relation E on a standard Borel space X is UBMC reducible to 2^{<ω_{1}} if there a UBMC function θ : X → 2^{<ω_{1}} such that for all x_{0}, x_{1} ∈ X

x_{0}Ex_{1} ⇔ θ(x_{0}) = θ(x_{1}).

Show that if E_{0} ≤_{B} E then E is not UBMC reducible to 2^{<ω_{1}}.

One can define a similar notion for functions from a space to the hereditarily countable sets. Thus we can say

θ : X → HC

is UBMC if whenever g : Y → X is Borel then there is a comeager set C ⊂ Y and Borel

y ↦ M_{y}
C → Mod(L),

where L is the language generated by a single binary relation E, such that for all y ∈ C

M_{y} ≅ ({θ(g(y))} ∪ TC(θ(g(y))), ∈);

in other words, on the comeager set C we have a Borel manner of supplying representatives of the hereditarily countable set θ(g(y)).

3.2. The definition of turbulence

Definition 3.13. Let G be a Polish group acting continuously on a Polish space X. The action is said to be turbulent if:

(i) every orbit is dense; (ii) every orbit is meager; (iii) for all x, y ∈ X, U ⊂ X, V ⊂ G open with x ∈ U, 1 ∈ V, there exists

y_{0} ∈ [y]_{G} and (g_{i})_{i∈ℕ} ⊂ V, (x_{i})_{i∈ℕ} ⊂ U with

x_{0} = x,
x_{i+1} = g_{i} · x_{i},

and for some subsequence (x_{n(i)})_{i∈ℕ} ⊂ (x_{i})_{i∈ℕ}

x_{n(i)} → y_{0}.

Part (i) in this definition is implied by (iii), but left in place to indicate the resemblance with 3.2. (i) and (ii) are already familiar from §3.1. Rather it is part (iii) which desires some comment.

Roughly speaking we might view condition (i) as saying that every orbit acts chaotically – a point in the space may begin at some specific place, but if we wait long enough the group action has the ability to rub it up arbitrarily close to any other orbit. Condition (iii) on the other hand states that not only do the orbits behave chaotically, but even when we try to restrict them and localize their behavior, they still have the capacity to approach representatives of any other orbit.

Some analogy might be provided by the old standby of a billiard ball bouncing back and forth on a perfectly smooth and frictionless table with perfectly elastic boundaries. Thinking of the position the ball as a function of time we may naturally associate this with an ℝ-action. Given reasonably generic starting conditions, we might expect that the ball will on its travels eventually pass through every part of the surface.

Now suppose we cordon off one section of the table by an array of one way doors. The ball may exit the cordoned section whenever moved to do so, but having done so it can never return. It we the examine the local orbit of the billiard ball inside this section we no longer expect the orbit to fill out the space. It should depart to never return after some finite time, and its entire local orbit consists of just finitely many finite line segments, and thus, since this is a compact 1-dimensional set, we may find a subregion of the cordoned section on which it has not had the chance to encroach.

Indeed it is true that the associated action of ℝ is not turbulent, and appropriately modified the above considerations can be turned into a proof that there are no turbulent ℝ-actions. The examples of turbulent actions in §3.3 all arise from infinite dimensional groups, and from 3.14 and 0.4, or the more general construction of §7.4 below, we can conclude that locally compact groups never give rise to turbulent actions. The notion of turbulence does not appear in the dynamics of locally compact groups, but rather is motivated by the general theory of equivalence relations and the much nastier groups which operate in this context.

We will prove that if E_{G} arises from a turbulent action and E_{S_{∞}} from a continuous S_{∞} action, then E_{G} is generically E_{S_{∞}}-ergodic. Before then there are some preliminaries, the first of which states that turbulent orbit equivalence relations are generically ergodic with respect to the equivalence relation on codes for countable subsets of 2^{ℕ}. In other words, any Baire measurable in the codes function from a turbulent Polish G-space to the countable subsets of 2^{ℕ} is constant on a comeager set.

Lemma 3.14. Let G be a Polish group acting continuously on a Polish space X with 3.13(iii): Namely: for all x, y ∈ X, U ⊂ X, V ⊂ G open with x ∈ U, 1 ∈ V, there exists y_{0} ∈ [y]_{G} and (g_{i})_{i∈ℕ} ⊂ V, (x_{i})_{i∈ℕ} ⊂ U with

x_{0} = x,
x_{i+1} = g_{i} · x_{i},

and for some subsequence (x_{n(i)})_{i∈ℕ} ⊂ (x_{i})_{i∈ℕ}

x_{n(i)} → y_{0}.

Then E_{G}, the orbit equivalence relation of G on X, is generically F_{2}-ergodic.

Prelude to the proof. As a preliminary to proving this lemma, let us first discuss the proof of the much easier result that if θ : X → B_{2} and various other relevant functions are all continuous with

∀x ∈ X∀g ∈ G(θ(x)F_{2}θ(g · x))

then x ↦ [θ(x)]_{F_{2}} is constant on X. Recall that B_{2} = {z ∈ 2^{ℕ×ℕ} : ∀k ≠ l(z(k, ·) ≠ z(l, ·)}. We have already seen at 2.30 that F_{2} ≤_{B} F_{2}|_{B_{2}}, and it is somewhat more natural to work on the set B_{2} where the equivalence relation F_{2} is actually induced by an S_{∞}-action. We fix x, y ∈ X and k ∈ ℕ and attempt to show how one might set about proving θ(x)(k, ·) ∈ {θ(y)(n, ·) : n ∈ ℕ}. Note that for any z in the space we may cover G by countably many sets {A_{l}^{z} : l ∈ ℕ} such that on each A_{l}^{z} we have for all g ∈ A_{l}^{z}

θ(g · z)(l, ·) = θ(z)(k, ·).

This is just at once true by the invariance assumption on θ.

Continuing with the policy of attempting to see how the proof moves forward under the most favorable circumstances, let us suppose that each such A_{l}^{z} is always open. Then for each z there will be some A_{l}^{z} which is not empty, and so we may find some z′ ∈ [z]_{G} and V a basic open neighborhood of the identity such that for all g ∈ V

θ(g · z′)(l, ·) = θ(z)(k, ·).

For each basic open V we may let B(V, l) be the set of z′ such that for all g ∈ V

θ(g · z′)(l, ·) = θ(z′)(l, ·).

Again, let us just optimistically assume that B(V, l) is always open.

Now going back to our original x and y, we will have some x′ ∈ [x]_{G} and basic open V containing the identity such that x′ ∈ B(V, l) and for all g ∈ V

θ(g · x′)(l, ·) = θ(x′)(l, ·) = θ(x)(k, ·).

Let U ⊂ B(V, l) be some basic open set containing x′. Using the assumption of turbulence applied to x′, y, U and V we may find y_{0} ∈ [y]_{G} and h_{i} ∈ V such that if we let x_{0} = x′ and x_{i+1} = h_{i} · x_{i} then y_{0} is an accumulation point of {x_{i} : i ∈ ℕ} and each x_{i} ∈ U ⊂ B(V, l).

At this point we can use definition of B(V, l) to note that each θ(x_{i+1})(l, ·) = θ(x_{i})(l, ·), and then obtain by induction i that each θ(x_{i})(l, ·) = θ(x)(k, ·). Thus by continuity of θ we have

θ(x)(k, ·) = θ(y_{0})(l, ·),

as required.

Of course this is all rather wishful. In general these sets will be far from being open. But we will steadily verify that the sets in question have the Baire property, and so there will be some open set which they equal on a comeager subset of X or G; these repeated verifications make the actual proof seem technical at first sight.

Proof. Choose θ : X → B_{2} ⊂ 2^{ℕ×ℕ} Borel such that for all x ∈ X and g ∈ G

{θ(x)(n, ·) : n ∈ ℕ} = {θ(g · x)(n, ·) : n ∈ ℕ}.

Let G_{0} < G be a countable dense subgroup, with (g_{i})_{i∈ℕ} some enumeration. Let (U_{n})_{n∈ℕ} be a countable basis of X and (V_{n})_{n∈ℕ} a neighborhood basis for G at the identity. Choose C ⊂ X a comeager Borel set on which θ is continuous by 2.49, and let C_{0} = ⋂_{i∈ℕ} g_{i} · C so that it is G_{0} invariant and then C_{1} = {x ∈ X : ∀^{∗}g ∈ G(g · x ∈ C_{0})}. By 2.47 C_{1} is comeager; by 2.50 Borel.

Claim(1): For all x ∈ C_{1} and k ∈ ℕ there exists W ⊂ G open and non-empty and l ∈ ℕ so that for all h ∈ W

h · x ∈ C_{0} ⇒ θ(x)(k, ·) = θ(h · x)(l, ·).

Proof of claim: Note that we may cover S_{∞} · θ(x) =_{df} [θ(x)]_{S_{∞}} by countably many closed sets (A_{j})_{j∈ℕ} so that for each j and y ∈ A_{j}

θ(x)(k, ·) = y(j, ·).

Let D_{x} be the preimage of C_{0} under g ↦ g · x and note that g ↦ θ(g · x) is continuous on D_{x}; since the preimage of each A_{j} is relatively closed and D_{x} is not meager, there must be some W ⊂ G non-empty and open so that for some j

∀g ∈ W ∩ D_{x}(θ(g · x) ∈ A_{j}),

and l = this j is as required. (□Claim(1))

Now define I : ℕ × X → ℕ ∪ {∞} so that I(l, x) equals the least n so that for all g_{i} and g_{i′} in V_{n}

g_{i} · x, g_{i′} · x ∈ C_{0} ⇒ θ(g_{i} · x)(l, ·) = θ(g_{i′} · x)(l, ·),

and equals ∞ if no such n exists. Note that if x ∈ C_{0} and I(l, x) = j then for all g ∈ V_{j}

g · x ∈ C_{0} ⇒ θ(g · x)(l, ·) = θ(x)(l, ·)

by the continuity of θ on C_{0} and the density of {g_{i} · x : i ∈ ℕ} in G · x ∩ C_{0}.

Considering ℕ ∪ {∞} as a discrete space, I is transparently Borel, so choose comeager C_{2} ⊂ C_{0} ∩ C_{1} on which I|_{ℕ×C_{2}} is continuous, and then take C_{3} = {x ∈ X : ∀^{∗}g ∈ G(g · x ∈ C_{2})}. Note C_{3} ⊂ C_{1}. Claim(1) implies that for all x ∈ C_{3} and k ∈ ℕ there will exist some x̂ ∈ [x]_{G} ∩ C_{2} and l for which I(l, x̂) < ∞ and θ(x)(k, ·) = θ(x̂)(l, ·). (We may assume that the h · x = x̂ promised by claim(1) is in C_{2}, since if not then we replace x̂ by ϵ · x̂ for some ϵ ∈ G which is close to the identity and has ϵ · x̂ ∈ C_{2}.)

With all this preparation let us finally suppose that x, y ∈ C_{3}; fixing k it is enough to show that θ(x)(k, ·) appears in {θ(y)(n, ·) : n ∈ ℕ}. By the above reasoning we can assume that x ∈ C_{2} and that

I(k, x) = m < ∞.

Since I is continuous on ℕ × C_{2} we can find some open set U ⊂ X containing x so that for all z ∈ C_{2} ∩ U

I(k, z) = m.

Choose an open neighborhood W of the identity in G so that W^{3} ⊂ V_{m} and let V = W ∩ W^{−1}.

Now apply the definition of turbulence to x, y, U, and V to find y_{0} ∈ [y]_{G} and {h_{i} : i ∈ ℕ} ⊂ V so that for x_{0} = x and x_{i+1} = h_{i} · x_{i} we have each x_{i} ∈ V and that y_{0} is an accumulation point of the set {x_{i} : i ∈ ℕ}.

Claim(2): There are {δ_{i} : i ∈ ℕ} ⊂ V so that d(x_{i}, δ_{i} ·x_{i}) < 2^{−i}, and δ_{i} ·x_{i} ∈ U, and δ_{i} · x_{i} ∈ C_{2}

Proof of claim: For each i let W_{i} ⊂ V be an open neighborhood of the identity such that for all δ ∈ W_{i}

d(x_{i}, δ · x_{i}) < 2^{−i}

and

δ · x_{i} ∈ U.

By definition of C_{3}, for each x_{i} there is a comeager subset of D_{i} ⊂ G on which g · x_{i} ∈ C_{2}. Choosing δ_{i} inside D_{i} ∩ W_{i} suffices. (□Claim(2))

Let x̂_{i} = δ_{i} · x_{i} for i > 0; x̂_{0} = x_{0}. Let ĥ_{i} = δ_{i+1}h_{i}δ_{i}^{−1} for i > 0; at i = 0 just let ĥ_{i} = δ_{1}h_{0}. Note that y_{0} is still an accumulation point of {x̂_{i} : i ∈ ℕ}.

Since each x̂_{i} ∈ C_{2}∩U we have I(k, x̂_{i}) = m for each i. Thus by ĥ_{i} ∈ W^{3} ⊂ V_{m}, the definition of I, and induction on i,

θ(x̂_{i})(k, ·) = θ(x̂_{0})(k, ·) =_{df} θ(x)(k, ·).

Claim(3): There is δ ∈ V_{m} such that δ · y_{0} ∈ C_{0} and at each i

δ · x̂_{i} ∈ C_{0},
θ(δ · x̂_{i})(k, ·) = θ(x̂_{i})(k, ·).

Proof of claim. The set of δ with δ · y_{0} ∈ C_{0} is comeager in G and therefore relatively comeager in V_{m} by y ∈ C_{1}, and similarly the set for which δ · x̂_{i} ∈ C_{0} is comeager at each i. However by the assumption that x̂_{i} ∈ U ∩ C_{2} we have that the set of δ ∈ V_{m} with θ(δ · x̂_{i})(k, ·) = θ(x̂_{i})(k, ·) is relatively comeager. Choosing δ in the intersection of all these comeager sets suffices. (□Claim(3))

By continuity of θ on C_{0},

θ(δ · y_{0})(k, ·) = lim_{i→∞}θ(δ · x̂_{i})(k, ·) = lim_{i→∞}θ(x̂_{i})(k, ·) = θ(x)(k, ·),

as required.

Definition 3.15. Let G be a Polish group, X a Polish G-space. For x ∈ X, U ⊂ X, V ⊂ G both open, with x ∈ U and 1_{G} ∈ V, we define the local U-V -orbit of x to be the set of x̂ for which there exists some k ∈ ℕ and x = x_{0}, x_{1}, x_{2}, ..., x_{k} ∈ U and g_{0}, g_{1}, ..., g_{k−1} ∈ V

x_{k} = x̂

and for all i < k

x_{i+1} = g_{i} · x_{i}.

This local orbit is denoted by O(x, U, V). We then define φ_{0}(x, U, V) to encode \overline{O(x, U, V)}, the closure of O(x, U, V), with the specification that if \{U_{l} : l ∈ ℕ\} is a basis of X then

φ_{0}(x, U, V) = {l ∈ ℕ : U_{l} ∩ O(x, U, V) ≠ ∅}.

Under the natural identification of 2^{ℕ} with the collection of all subsets of ℕ the object φ_{0}(x, U, V) can be calculated in a Borel fashion from x, in the sense that

φ_{0}(·, U, V) : X → 2^{ℕ}

x ↦ {l : ∃k∃(x_{i})_{i≤k} ⊂ U∃(g_{i})_{i<k} ⊂ V (x_{0} = x, x_{k} ∈ U_{l}, ∀i < k(x_{i+1} = g_{i} · x_{i}))} is a Borel function for each open U and V: For each k let A_{k}(U, V, U_{l}) be the set of x for which there exist open U′_{0}, U′_{1}, ...U′_{k} = U_{l} ∩ U, and g_{0}, ..., g_{k−1} ∈ V with x ∈ U′_{0}, g_{i} · (U′_{i}) ⊂ U′_{i+1}; each A_{k}(U, V, U_{l}) is itself open, and l ∈ φ_{0}(x, U, V) if and only if x ∈ A_{k}(U, V, U_{l}) for some k.

With the definition of 3.15 we can express 3.13(iii) compactly: for all x ∈ U ⊂ X and 1_{G} ∈ V ⊂ G

[y]_{G} ∩ \overline{O(x, U, V)} ≠ ∅.

Lemma 3.16. Let G be a Polish group and X a Polish G-space, and suppose that E_{G}^{X} is generically F_{2}-ergodic. Then for all U ⊂ X open, V ⊂ G open containing the identity there is a comeager C ⊂ X so that for all x ∈ C and x_{0} ∈ [x]_{G} ∩ U the set \overline{O(x_{0}, U, V)} has non-empty interior.

Proof. Let (g_{i})_{i∈ℕ} enumerate a countable dense subset of G. Let W be the interior of the complement of U.

First note that for any closed nowhere dense A and i ∈ ℕ the set

B_{i}(A) = {x ∈ X : g_{i} · x ∈ W } ∪ {x ∈ X : g_{i} · x ∈ U \ A}

is open dense, and so

B(A) = ⋂_{i∈ℕ} B_{i}(A)

is a dense G_{δ}.

Using the assumptions of the lemma we may go to a comeager set C on which

x ↦ {φ_{0}(g_{i} · x, U, V) : g_{i} · x ∈ U} = {φ_{0}(x_{0}, U, V) : x_{0} ∈ [x]_{G} ∩ U}

is constant. And now for any x ∈ C and A ∈ \{\overline{O(g_{i} · x, U, V)} : g_{i} · x ∈ U\} it must be the case that A contains an open set, or else we find y ∈ C ∩ B(A) for which A will not be an element of \{\overline{O(g_{i} · y, U, V)} : g_{i} · y ∈ U\}.

If X is a Polish G-space with (i) and (ii) from the definition of turbulence, and the conclusion of 3.16 holds for every x ∈ X then the action is turbulent.

Lemma 3.17. Let G and H be Polish groups and X and Y Polish G and Hspaces respectively. Suppose that

θ : X → Y

is a Borel function such that for all x ∈ X and g ∈ G

θ(x)E_{H}θ(g · x).

Then for any open neighborhood W of the identity in H there is a comeager set of x ∈ X for which there is a basic open neighborhood V of the identity in G with

∀^{∗}g ∈ V ∃h ∈ W (h · θ(x) = θ(g · x)).

Proof. First:

Claim: For all x ∈ X, there is a comeager set of g_{0} ∈ G for which there exists some open neighborhood V of the identity on which

∀^{∗}g_{1} ∈ V ∃h ∈ W (h · θ(g_{0} · x) = θ(g_{1}g_{0} · x)).

Proof of claim: Fix x and W. Choose a smaller open

(W′)^{2} ⊂ W
1_{H} ∈ W′
(W′)^{−1} = W′.

We may cover H by countably many right translates {W′h_{i} : i ∈ ℕ}, for appropriately chosen (h_{i})_{i∈ℕ}. Therefore we may cover

θ[[x]_{G}] =_{df} {θ(x_{0}) : x_{0} ∈ [x]_{G}}

by {W′h_{i} · θ(x) : i ∈ ℕ}. Each W′h_{i} · θ(x) is Σ ∼_{1}^{1}, and thus its pullback under the function

g ↦ θ(g · x)

is Σ ∼_{1}^{1}, and so has the Baire property by 2.53. So find open (O_{i})_{i∈ℕ} with ⋃_{ℕ} O_{i} open dense in G and for all i ∈ ℕ

∀^{∗}g ∈ O_{i}(θ(g · x) ∈ W′h_{i} · θ(x)).

At each i let C_{i} ⊂ O_{i} be this relatively comeager set; ⋃_{i∈ℕ} C_{i} is comeager in G, so consider any i and choose g_{0} ∈ C_{i}. Take V to be a basic open neighborhood of the identity for which we have V g_{0} ⊂ O_{i}. C_{i}(g_{0})^{−1} is still relatively comeager in V, and for all g_{1} ∈ C_{i} ∩ C_{i}(g_{0})^{−1} ∩ V we have some h ∈ W′ with

θ(g_{0} · x) = hh_{i} · θ(x);

since g_{1}g_{0} ∈ C_{i} there is some h′ ∈ W′ with

θ(g_{1}g_{0} · x) = h′h_{i} · θ(x);

and so for h^{∗} ∈ W′(W′)^{−1} ⊂ W given by h^{∗} = h′h^{−1} we have

h^{∗} · θ(g_{0} · x) = θ(g_{1}g_{0} · x).

Thus for a comeager set of g_{1} in V we do indeed have the conclusion of the claim. (□Claim)

The set of x for which the conclusion of the lemma holds is Σ ∼_{1}^{1} by 2.55, 2.61(ii) and hence has the Baire property by 2.53 again. If the lemma fails then for all g_{0} ∈ G we would have that on a non-meager set of x ∈ X it is the case that for all open neighborhoods V of the identity,

∃^{∗}g_{1} ∈ V ∀h ∈ W (h · θ(g_{0} · x) ≠ θ(g_{1}g_{0} · x)).

By Kuratowski-Ulam as at 2.46, there is consequently a non-meager set of x ∈ X and a non-meager set of g_{0} ∈ G such that for all open V containing 1_{G}

∃^{∗}g_{1} ∈ V ∀h ∈ W (h · θ(g_{0} · x) ≠ θ(g_{1}g_{0} · x)),

with a contradiction to the claim.

Theorem 3.18. Let G be a Polish group, X a Polish G-space, satisfying (iii) from the definition of turbulence: For all x, y ∈ X, all U ⊂ X open containing x, all V ⊂ G open containing 1_{G},

\overline{O(x, U, V)} ∩ [y]_{G} ≠ ∅.

Let Y be a Polish S_{∞}-space.

Then E_{G}^{X} is generically E_{S_{∞}}^{Y}- ergodic.

Proof. Let θ : X → Y be a Borel map such that x_{1}E_{G}^{X}x_{2} ⇒ θ(x_{1})E_{S_{∞}}^{Y}θ(x_{2}). Let (U_{l})_{l∈ℕ} be a basis of X and (V_{l})_{l∈ℕ} a neighborhood basis of G at the

identity. Define I : X × ℕ → ℕ ∪ {∞} by setting I(x, N) to be the least l so that

∀^{∗}g ∈ V_{l}∃h ∈ S_{∞}(∀i ≤ N(h(i) = i) ∧ h · θ(x) = θ(g · x)),

if such l exists (= ∞ if not). If ℕ ∪ {∞} is given the discrete topology then the preimage of a basic open set is in the σ-algebra generated by Σ ∼_{1}^{1} sets.

Using lemmas 2.49 and 3.17, let C_{0} be a comeager set on which I and θ are continuous and that for all x ∈ C_{0}, N ∈ ℕ

I(x, N) < ∞.

Following 3.16 find C_{1} ⊂ X comeager and invariant so that for all l, k ∈ ℕ and x ∈ C_{1} ∩ U_{l} the closure of the local U_{l}-V_{k}-orbit,

\overline{O(x, U_{l}, V_{k})},

contains an open set. Now let

C_{2} = {x ∈ C_{0} : ∀^{∗}g ∈ G(g · x ∈ C_{0} ∩ C_{1})};

by 2.47 this is comeager. Fix x, y ∈ C_{2}. We will see that there exists g, h ∈ S_{∞} with g · θ(x) = h · θ(y), and thereby complete the proof.

We choose successively

x_{0} = x, x_{1}, x_{2}, ...,
y_{0} = y, y_{1}, y_{2}, ...,

in C_{2}, and

g_{0} = 1_{S_{∞}}, g_{1}, g_{2}...,
h_{0} = 1_{S_{∞}}, h_{1}, h_{2}, ...,

in S_{∞}, and

W_{0} = X, W_{1}, ..., W′_{0}, W′_{1}, ...,

basic open in X,

V_{n(0)}, V_{n(1)}, ..., V_{k(0)}, V_{k(1)}, ...,

basic open neighborhoods of 1_{G} so that at each i (ia) g_{i} · θ(x) = θ(x_{i}); (ib) h_{i} · θ(y) = θ(y_{i}); (iia) x_{i+1} ∈ W_{i+1} ∩ C_{2} ∩ O(x_{i}, W_{i}, V_{n(i)}); (iib) y_{i+1} ∈ W′_{i+1} ∩ C_{2} ∩ O(y_{i}, W′_{i}, V_{k(i)}); (iii) W_{i} ⊃ W′_{i} ⊃ W_{i+1}; (iv) for d a previously fixed complete metric on Y and i > 0

d(θ[W_{i} ∩ C_{2}]) < 2^{−i};

(va) for

N_{i+1} =_{df} sup{g_{i+1}(l), g_{i+1}^{−1}(l) : l ≤ i + 1}

we have

(W_{i+1} ∩ C_{2}) × {N_{i+1}} ⊂ I^{−1}[{n(i + 1)}];

(vb) for

K_{i} =_{df} sup{h_{i}(l), h_{i}^{−1}(l) : l ≤ i}

we have

(W′_{i} ∩ C_{2}) × {K_{i}} ⊂ I^{−1}[{k(i)}];

(via) for all l ≤ i + 1 < j + 1

g_{j+1}(l) = g_{i+1}(l)
g_{j+1}^{−1}(l) = g_{i+1}^{−1}(l);

(vib) for all l ≤ i < j

h_{j}(l) = h_{i}(l)
h_{j}^{−1}(l) = h_{i}^{−1}(l);

(viia) O(x_{i}, W_{i}, V_{n(i)}) is dense in W′_{i}; (viib) O(y_{i}, W′_{i}, V_{k(i)}) is dense in W_{i+1}.

Note that if all this succeeds we may in the end let

g_{∞} = lim_{i∈ℕ}g_{i}

and

h_{∞} = lim_{i∈ℕ}h_{i};

by (vi) the limits exist. Then

g_{∞} · θ(x) = h_{∞} · θ(y)

by (ii)-(iv).

We can begin the construction with x_{0} = x, W_{0} = X, V_{n(0)} = G, g_{0} = 1_{S_{∞}}. Then choose y_{0} = y, W′_{0} containing y with

(W′_{0} ∩ C_{2}) × {0} ⊂ I^{−1}[{k(0)}],

some k(0) ∈ ℕ, and let h_{0} = 1_{S_{∞}}.

So let us suppose we have built these as above for all j ≤ i and concentrate on producing x_{i+1}, g_{i+1}, and W_{i+1}. The further step of producing y_{i+1}, h_{i+1}, and W′_{i+1} is completely symmetrical and can be omitted.

Fix nonempty open U ⊂ W′_{i} in which O(y_{i}, W′_{i}, V_{k(i)}) is dense and then also demand that d(θ[U ∩ C_{2}]) < 2^{−i−1}. Then choose x_{i+1} ∈ O(x_{i}, W_{i}, V_{n(i)}) ∩ U, and let z_{0} = x_{i}, z_{1}, ..., z_{k} ∈ W_{i} and g′_{0}, g′_{1}, ..., g′_{k−1} ⊂ V_{n(i)} witness this, in the sense that

W_{i} O(x_{i}, W_{i}, V_{n(i)}) W′_{i} O(y_{i}, W_{i}, V_{n(i)}) W_{i+1} •x_{i} •y_{i} •x_{i+1}

Figure 3.1. The local orbits folding in

z_{i+1} = g′_{i} · z_{i}, z_{0} = x_{i}, z_{k} = x_{i+1}. From now on we may as well assume that i > 0, or else (via) places no restriction on our choice of g_{1}.

Claim(I): Without loss of generality each z_{j} ∈ C_{2}.

Proof of claim: Choose V′_{j} open neighborhoods of the identity of G so that V′_{j} = (V′_{j})^{−1}, V′_{j+1} ⊂ V′_{j}, and

V′_{j}g′_{j}V′_{j} ⊂ V_{n(i)}.

Since

∀^{∗}g ∈ G(g · x_{i} ∈ C_{2})

and each z_{j} ∈ [x]_{G} we may find ϵ_{j} ∈ V′_{j} so that

ϵ_{j} · z_{j} ∈ C_{2} ∩ W_{i}.

If we now replace

z_{0}, z_{1}, z_{2}, ..., z_{k}

by

z_{0}, ϵ_{1} · z_{1}, ϵ_{2} · z_{2}, ..., ϵ_{k} · z_{k},

and

x_{i+1}

by

ϵ_{k} · x_{i+1},

and

g′_{0}, g′_{1}, ..., g′_{k−1}

by

ϵ_{1}g′_{0}, ϵ_{2}g′_{1}(ϵ_{1})^{−1}, ϵ_{3}g′_{2}(ϵ_{2})^{−1}, ..., ϵ_{k}g′_{k−1}(ϵ_{k−1})^{−1}

then we are done. (□Claim(I))

Thus in particular for i > 0 I(z_{j}, N_{i}) = n(i) for each j. With slight perturbation of the z_{j}’s we can use the assumption

I(z_{j}, N_{i}) = n(i)

as follows:

Claim(II): Without loss of generality for each j there is f_{j} ∈ S_{∞} with

f_{j} · θ(z_{j}) = θ(z_{j+1})
∀l ≤ N_{i}(f_{j}(l) = l).

Proof. First fix a relatively comeager B(z_{0}) ⊂ V_{n(i)} so that ∀g ∈ B(z_{0}) we have that there exists f_{0} ∈ S_{∞} with

f_{0} · θ(z_{0}) = θ(g · z_{0}),
∀l ≤ N_{i}(f_{0}(l) = l).

This uses I(z_{0}, N_{i}) = n(i). Then by the argument of claim(I) we may replace g′_{0} by some ϵ · g′_{0} ∈ B(z_{0}) and then by the definition of B(z_{0}) find f_{0} as desired. Clearly this can be continued inductively so that each g′_{j} ∈ B(z_{j}), for B(z_{j}) defined by the requirement that it be relatively comeager and so that ∀g ∈ B(z_{j}) we have that there exists f_{j} ∈ S_{∞} with f_{j} · θ(z_{j}) = θ(g · z_{j}) and ∀l ≤ N_{i}(f_{j}(l) = l).

(□Claim(II))

Let f_{i}^{∗} = f_{k−1}f_{k−2}...f_{0} and note then θ(z_{k}) =_{df} θ(x_{i+1}) = f_{i}^{∗} · θ(x_{i}). Each f_{j} leaves 0, 1, ..., N_{i} fixed, and hence so does f_{i}^{∗}; then we may let g_{i+1} = f_{i}^{∗}g_{i}, which indeed gives us (va) above for g_{i+1}. Now take basic non-empty open W_{i+1} ⊂ U containing x_{i+1} and some n(i + 1) so that

(W_{i+1} ∩ C_{2}) × {N_{i+1}} ⊂ I^{−1}[{n(i + 1)}],

where as indicated above N_{i+1} is the sup of {g_{i+1}(l), g_{i+1}^{−1}(l) : l ≤ i}.

Corollary 3.19. (A) If G is a Polish group and X is a turbulent Polish Gspace, then for any Polish S_{∞}-space Y

E_{G}^{X} ≰_{B} E_{S_{∞}}^{Y}.

(B) No turbulent orbit equivalence relation admits classification by countable models.

Also following from the proof of 3.18 is that there can be no Baire measurable reduction of a turbulent orbit equivalence relation to any E_{S_{∞}}^{Y}. Since these arguments were based on Baire category methods it is natural to relax the definitions to the requirement that they hold on a comeager set.

Definition 3.20. Let G be a Polish group and X a Polish G-space. The action is said to be generically turbulent if:

(i) ∀^{∗}x ∈ X([x]_{G} is dense); (ii) every orbit is meager;

(iii) ∀^{∗}x, y ∈ X, for all U ⊂ X and V ⊂ G basic open with x ∈ U and V containing the identity, there exists y_{0} ∈ [y]_{G} with

y_{0} ∈ \overline{O(x, U, V)}.

Theorem 3.21. Let G be a Polish group, X a Polish G-space. Then the following are equivalent:

(I) The action is generically turbulent.

(II) The orbit equivalence relation E_{G}^{X} is generically F_{2}-ergodic.

(III) For any countable language L, the orbit equivalence relation is generically ≅|_{Mod(L)}-ergodic.

(IV) Every orbit is meager; for a comeager collection of x ∈ X the orbit [x]_{G} is dense and for all U ⊂ X, V ⊂ G basic open with x ∈ U, 1 ∈ V, there exists U_{0} ⊂ U non-empty with O(x, U, V) dense in U_{0}.

(V) Every orbit is meager, and for a comeager collection of x ∈ X it is the case that [x]_{G} is dense and for all open U ⊂ X, V ⊂ G with x ∈ U, 1 ∈ V, there exists U_{0} ⊂ U non-empty with O(x, U, V) dense in U_{0}.

(VI) Some orbit is dense, every orbit is meager, and there exists x ∈ X, such that for all U ⊂ X, V ⊂ G basic open with x ∈ U, 1 ∈ V, there exists open U_{0} ⊂ U non-empty with O(x, U, V) dense in U_{0}.

(VII) There is an invariant dense G_{δ} subspace X_{0} ⊂ X with X_{0} a turbulent Polish G-space.

Proof. (I)⇔(II)⇔(III)⇔(IV): (I)⇒(II) follows exactly as in 3.14, and (II)⇒(III) is as in 3.18, while (III)⇒(II) is trivial from 2.41. (II)⇒(IV) is the proof of 3.16. (IV)⇒(I): since if

U_{0} ⊂ \overline{O(x, U, V)}

is open and non-empty and if [y]_{G} is dense, then we may find y_{0} ∈ [y]_{G} ∩\overline{O(x, U, V)}, as desired.

(IV)⇔(V): It is immediate that (V) entails (IV). As for the converse, if x satisfies the statement of (IV), with its limitation to basic open sets, and U and V are as in (V), then we may find basic open U′ ⊂ U and V′ ⊂ V containing x and 1 respectively; noting that O(x, U′, V′) ⊂ O(x, U, V) we are done by assumption that O(x, U′, V′) is somewhere dense.

So we have the equivalence of (I)-(V) and we just need to tie these to (VI) and (VII).

Trivially (IV)⇒(VI).

(VI)⇒(IV): For this purpose let (g_{i})_{i∈ℕ} be a dense countable subset of G, {U_{l} : l ∈ ℕ} a countable basis of X, {V_{l} : l ∈ ℕ} a countable neighborhood basis at 1_{G}, and let x ∈ X witness the conclusion of (VI).

Now for each g_{i} ·x ∈ U_{l} and V_{k} let W_{l,i,k} ⊂ U_{l} be the interior of \overline{O(g_{i} · x, U_{l}, V_{k})}. This set will be non-empty by assumption on [x]_{G}.

Note that [x]_{G} must be dense, since if we choose some [z]_{G} which is dense, then we may find z′ ∈ W_{l,i,k} ∩ [z]_{G} – and since z′ is a limit point of [x]_{G} and since [z′]_{G} = [z]_{G} is dense, [x]_{G} is dense as well.

Let B_{l,k} be the set of y ∈ X such that either y is not in U_{l} or there exists some i with

O(y, U_{l}, V_{k}) ∩ W_{l,i,k} ≠ ∅.

Note that each B_{l,k} is G_{δ} by direct calculation, and dense since [x]_{G} ⊂ B_{l,k}. For V_{m} ∩ W_{l,i,k} ≠ 0 and g_{i} · x ∈ U_{l}, let C_{l,i,k,m} be the set of y ∈ W_{l,i,k} for which

O(y, U_{l}, V_{k}) ∩ V_{m} ≠ ∅.

C_{l,i,k,m} is open and dense in W_{l,i,k} whenever g_{i} · x ∈ U_{l} since

W_{l,i,k} ∩ O(g_{i} · x, U_{l}, V_{k}) ⊂ C_{l,i,k,m}.

Hence if we let D_{l,i,k} be the set of y ∈ Y such that

∀m((y ∈ W_{l,i,k} ∧ (U_{m} ∩ W_{l,i,k} ≠ ∅)) ⇒ O(y, U_{l}, V_{k}) ∩ V_{m} ≠ ∅)

then D_{l,i,k} is a dense G_{δ}.

Now for any y with [y]_{G} dense and in the intersection of all the g_{j} · B_{l,k} and g_{j} · D_{l,i,k} we will have that the local orbits of y are all somewhere dense. By 3.4 and the above calculations, the set of such y is comeager.

(VII)⇒(IV): At once since the conclusion of (IV) must hold on X_{0} by 3.16, and therefore X since X_{0} is comeager in X.

For (IV)⇒(VII): Following the notation of (VI)⇒(IV) let X_{0} be the set

( ⋂_{i,l,k}(B_{l,k} ∩ D_{l,i,k}))^{∗G}.

Exercise 3.22. If X is a turbulent Polish G-space, then there is no Σ ∼_{1}^{1} set A ⊂ X meeting every orbit in a countable but non-empty set. (Use 2.54.)

3.3. Examples

Example 3.23. Recalling c_{0} = {x⃗ ∈ ℝ^{ℕ} : x_{i} → 0} equipped with the topology given by the sup norm, d(x⃗, y⃗) = sup_{i∈ℕ}|x_{i} − y_{i}|, we may let c_{0} act on the ambient space ℝ^{ℕ} by translation. The inclusion map c_{0} ↪ ℝ^{ℕ} is continuous and so this makes ℝ^{ℕ} a Polish c_{0}-space. This is one of the simplest examples of a turbulent action.

It is clear that every orbit is dense, since c_{0} is dense. We obtain every orbit being meager from c_{0} being meager, which in turn follows by 2.56(i) and (−c_{0}) + c_{0} = c_{0} not containing an open set. As for the final condition 3.13(iii) to the effect that every orbit is somehow “locally dense”, fix x⃗, y⃗ ∈ ℝ^{ℕ}, U ⊂ ℝ^{ℕ} containing x⃗, V ⊂ c_{0} an open neighborhood of the identity in c_{0}. We may assume that for some ϵ > 0, l ∈ ℕ

V = {z⃗ ∈ c_{0} : sup_{i∈ℕ} z_{i} < ϵ}
U = {z⃗ ∈ ℝ^{ℕ} : ∀i < l(|z_{i} − x_{i}| < ϵ)},

and that y⃗ ∈ U. Fixing some arbitrary open neighborhood U_{0} of y⃗ it suffices to find some z⃗ ∈ O(x, U, V) ∩ U_{0}.

So using the density of [x]_{c_{0}} we may find g⃗ ∈ c_{0} with g⃗ · x ∈ U_{0} ∩ U. Let δ = sup_{ℕ}|g_{i}|, and choose k ∈ ℕ large enough that δ/k < ϵ. Let h⃗ = g⃗/k, in the sense that h_{i} = g_{i}/k at each i ∈ ℕ. Let h⃗^{j} = h⃗ for each j < k, and

x⃗_{0} = x⃗,
x^{j+1} = h⃗^{j} · x^{j}.

Then x^{k} ∈ U_{0} and each h⃗^{j} ∈ V.

A similar result holds for l^{1}(ℝ) = {x⃗ ∈ ℝ^{ℕ} : ∑ |x_{i}| < ∞} or l^{1}(ℂ) = {x⃗ ∈ ℂ^{ℕ} : ∑ |x_{i}| < ∞}. There is a completely general result regarding Polishable subgroups of ℝ^{ℕ}.

Definition 3.24. Let G ⊂ ℝ^{ℕ} be a Polishable subgroup. G is said to be super dense if for all (x_{0}, x_{1}, ...x_{n}) ∈ ℝ^{<ℕ} there is some y⃗ ∈ G with y_{i} = x_{i} for all i ≤ n.

Proposition 3.25. Let G ⊂ ℝ^{ℕ}, G ≠ ℝ^{ℕ} be super dense. Then the action by translation of G on ℝ^{ℕ} is turbulent.

Proof. Let τ be the Polish topology on G. Note that by 2.56

π_{n} : G → ℝ^{n}
x⃗ ↦ (x_{0}, x_{1}, ..., x_{n−1})

is a continuous map; by assumption of super density it is onto and so open.

Now we can simply repeat the proof from 3.23 with suitable changes. The argument that every orbit is dense and meager is as before.

So fix U ⊂ ℝ^{ℕ} open containing x⃗, and assume that U = {z⃗ ∈ ℝ^{ℕ} : ∀i < l(|z_{i} − x_{i}| < ϵ)}. Fix y⃗ ∈ ℝ^{ℕ}, and note that we may find some g⃗ ∈ G with g⃗ · y⃗ ∈ U by assumption on G. Let U_{0} be an open set containing g · y⃗ and V an open set containing 1_{G}. We may assume

U_{0} = {z⃗ ∈ ℝ^{ℕ} : ∀i < n(|z_{i} − y_{i}| < ϵ_{0})}.

Then we may find some (h_{0}, h_{1}, ...) ∈ ℝ^{n} so that h_{i} · x_{i} = y_{i}.

Since π_{n} is an open map we can find some open W ⊂ ℝ^{n} with π_{n}[V] = W. Then for some sufficiently large k we will have (h_{0}, h_{1}, ...h_{n−1})/k ∈ W. Choosing g⃗ ∈ V with g_{i} = h_{i}/k for i < n, we may let x^{0} = x⃗, x^{j+1} = g⃗ · x^{j}.

Examples 3.26. Under the natural identification of 2^{ℕ} with ℤ_{2}^{ℕ} we may view the Cantor space as a compact Polish group with the operation of pointwise addition. A further identification is obtained by associating every f ∈ 2^{ℕ} with the set A_{f} ∈ P(ℕ) (the collection of all subsets of ℕ) defined by

n ∈ A_{f} ⇔ f(n) = 1.

Under this identification the group operation on 2^{ℕ} corresponds in P(ℕ) to the operation of symmetric difference:– AΔB = A \ B ∪ B \ A.

A subset of P(ℕ) is said to be an ideal if closed under finite unions and the process of going to an arbitrary subset. An ideal on P(ℕ) is said to be Polishable if it is Polishable as a group with respect to the operation of symmetric difference.

Two examples of Polishable ideals on P(ℕ) are: (i) the density ideal:

I_{d} = \{A ∈ P(ℕ) : \frac{|A ∩ \{0, 1, .., n\}|}{n + 1} → 0\};^{1}

(ii) the summable ideal:

I_{2} = \{A ∈ P(ℕ) : ∑_{n∈A} \frac{1}{n + 1} < ∞\}.

These groups are Polish in the topologies generated by the metrics

d_{0}(A, B) = sup_{n∈ℕ} \frac{|AΔB ∩ \{0, 1, .., n\}|}{n + 1}

1Where |A ∩ {0, 1, .., n}| denotes the size of the set A ∩ {0, 1, .., n}.

and

d_{1}(A, B) = ∑_{n∈AΔB} \frac{1}{n + 1}.

Both the actions by translation on the surrounding space 2^{ℕ} are turbulent. This also provides a counterpoint to the examples of 3.25, since here O(x, U, V) depends as much on V as it does on U – for instance in (i) (or (ii)) above, if we let U = P(ℕ), A be any element in P(ℕ) and

V_{k} = {B ∈ I_{2} : ∀n < k(n ∉ B)}

then

\overline{O(A, U, V_{k})} = \{C ⊂ ℕ : ∀n < k(n ∈ C ⇔ n ∈ A)\}.

More recently Alexander Kechris has in [54] used the notion of turbulence and work of Slawek Solecki from [77] to completely characterize for which Borel ideals I ⊂ P(ℕ) we have the equivalence relation E_{I} given by

AE_{I}B ⇔ AΔB ∈ I

admitting classification by countable models.

A rather different kind of example for a group studied in functional analysis:

Theorem 3.27. (Kechris, Sofronidis) The action of U_{∞} upon itself by conjugation is generically turbulent.

[10] shows the conjugation action of U_{∞} to be meager and to have a dense orbit, but the condition (iii) from turbulence did not follow from this almost classical argument. Kechris and Sofronidis not only obtained (iii) but determined the orbits x ∈ U_{∞} satisfying condition (V) from 3.21.

We will investigate a sequence of examples similar to this one in chapters 4 and 5.

Example 3.28. Choose a rapidly increasing sequence of primes, p_{0}, p_{1}, ..., so that p_{k+1} > kp_{0}p_{1}...p_{k}. Let G = ℤ_{p_{0}} × ℤ_{p_{1}} × ℤ_{p_{2}}..., so the elements of G will be functions f : ℕ → ℕ such that f(n) < p_{n} and addition is mod p_{n} on the nth coordinate. Then let

H = \{f ∈ G : sup_{ℕ} \frac{|f(n)|_{n}}{p_{n}} \} < ∞,

where |f(n)|_{n} should be understood as indicating the least m ≥ 0 for which f(n) + m = p_{n} or f(n) − m = 0. H is a Polish group with respect to this sup norm and the action of H on G is turbulent.

In some respects this example is one of minimal algebraic complexity. Not only is the group H abelian, but also monothetic – the group element 1 ∈ H given by 1(n) = ¯1 for all n generates a dense cyclic subgroup of H.

Example 3.29. Finally Kechris has found an application in [55] which would initially appear to have no relation to the study of equivalence relations. Here he unearthed a completely new proof of a result of Preiss and Rataj to the effect that there is no analytic maximal set of pairwise orthogonal measures on any uncountable standard Borel space; in fact, this result was not only given a faster proof but considerably strengthened. Turbulence found an entrance by Kechris’ embedding of a turbulent orbit equivalence relation into the equivalence relation of absolute continuity of measures, to which he then applied a version of 3.22.

3.4. Historical remarks

3.4.1. Smoothness as a notion of classification. The results of §3.1 are implicit in classical papers. Indeed some version of these results have appeared independently in many different places – for instance [25] and [5], and in [47] one may find a discussion of even earlier proofs that ℝ/ℚ is not smooth.

Smoothness is very commonly taken to be archetypal notion of classifiable. In the writings of some analysts, such as perhaps [18], one almost senses a feeling that smoothness might be the only thing that could be meant by providing a complete classification of an equivalence relation; that somehow this is the only sensible way one could even think about the problem. The interesting properties are those that can be presented as Borel subsets of a Polish space. A classification theorem for an equivalence relation E on a space X should consist in finding a family of invariant, preferably Borel invariant, subsets of X, and the classification is complete when any two distinct equivalence classes can be separated by a member of our family. In order for the notion of completely classifiable to have any content, and not just apply across the board to every virtually every equivalence relation, we should require that the family of invariant Borel sets be countable. Thus by 2.27(ii) we are inexorably led to the conclusion that an equivalence relation is completely classifiable if and only if it is smooth.

This is also the view point the authors of [25] and [14] have drawn from [63], and despite its plausibility it turns out to be a surprisingly harsh standard. It is a position from which we must conclude that even the rank one torsion free abelian groups are unclassifiable and must reject the K-theoretic classification of [24]; any equivalence relation into which we may embed E_{0} is by this definition unclassifiable.^{2}

On the other hand a completely different culture is ascendant in the branch of mathematical logic known as model theory. For instance one finds implicit in the introductory passages of [2] that the classification of countable structures is completely and satisfactorily settled by the Scott analysis, which we will recall at §6.1 below, and that the crucial directions of research are related to which classes of uncountable, discrete structures allow some higher level analogue of Scott’s original analysis. From this standing it is simply a triviality that all the finite rank, or even infinite rank, torsion free abelian groups can be completely classified.

This book has tried to take from both these points of view. §3.1 recalls a coherent theory of smoothness, and the work of §3.2 came about as an attempt to parallel these classical and folklore results to more permissive notions of classification, such as the assignment of a countable structure considered up to isomorphism as a complete invariant.

2A footnote is irresistible . The perspective of [21] is to indeed embrace rank one torsion free abelian groups as classifiable since we may associate to them elements of P(ℕ)/Finite as complete invariants – in fact they are considered classifiable precisely because ≅|_{TFA_{1}} ≤_{B} E_{0}. Rather [21] seems to take the battle ground to be for rank two torsion free abelian groups.

Recently [36] showed that for all n ≥ 2 one has E_{0} <_{B} ≅|_{TFA_{n}}; the argument used the non-amenability of SL_{2}(ℤ) for rank 2 and Kazhdan’s property (T) for n ≥ 3 to obtain not only E_{0} <_{B} ≅|_{TFA_{n}} but also the non-treeability of ≅|_{TFA_{n}} – which in essence states that ≅|_{TFA_{n}} is not ≤_{B} reducible to any free Borel action of a countable free group. Since then [56] obtained the parallel non-treeability result for n = 2 through an appeal to deep facts about algebraic groups.

The exact classification difficulty of the ≅|_{TFA_{n}}’s remains undetermined, but at the time of writing it would appear intertwined with the general ergodic theory of countable group actions.

3.4.2. Alternative proofs. In particular §3.2 and §3.3 show that there are Polish group actions whose orbit equivalence relations are not reducible to orbit equivalence relations induced by S_{∞}. This question was raised at 3.5.4 and 8.2.4 of [4]. As a comment on the viewpoint at the time of posing, almost all the Polish group actions which had been seriously studied were so reducible; most examples of orbit equivalence relations commonly considered bore a resemblance to the logic action of S_{∞} on Mod(L) or the actions of locally compact groups.

During 1996 the perspective began to change. Groups arising from real analysis such as the classical Banach spaces were subjected to increasing scrutiny, and the counterexamples that once seemed exotic became commonplace.

It should not however be suggested that 3.18 was the first demonstration that there are such unclassifiable group actions. Earlier in spring of 1995 the author had a rather different construction, using work of [77] to the effect that there are F_{σ} Polishable ideals whose coset equivalence relation is not reducible to a countable equivalence relation and the analysis in [38] of essentially F_{σ} equivalence relations induced by S_{∞} all being reducible to countable equivalence relations. The broader results of §3.2 came later.

Indeed even after the development of turbulence, 3.18 was not the first proof of non-reducibility to S_{∞}-actions. Earlier it was shown by a transfinite induction along the rank of the corresponding Scott sentence associated to θ(x) that for any generically F_{2}-ergodic E we must in general further obtain that E is generically E_{S_{∞}}^{Y}-ergodic. This alternative proof was not only shorter, but also more informative. Its disadvantage was in being less than completely elementary. Moreover the proof presented at 3.18 turns out to be useful in the construction from §9.2.

All well and good. Later an offhand reference was noticed in [20] to an unpublished proof of the density ideal’s coset equivalence relation not admitting classification by countable models. Trivially this ideal is Polishable, and thus Harvey Friedman and Lee Stanley answered a principal question of [4] before it was ever even asked.

In early 1997 Friedman visited Los Angeles and made available the argument, which should eventually be published in [19]. In fact it was indeed very close to the afore mentioned induction along the Scott sentence.

3.4.3. Borel dichotomy theorems. A towering influence has been exerted on the theory of Borel equivalence relations by [30]:

Theorem 3.30. (Harrington, Kechris, Louveau) Let E be a Borel equivalence relation. Then exactly one of the following two things must happen:

1. E ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E.

This is the paradigm dichotomy theorem: Either an equivalence relation is smooth, or its non-smoothness is explained by the embeddability of E_{0}. The result fails for general Σ ∼_{1}^{1} equivalence relations, even those induced by the continuous actions of Polish groups; but in [37] it is noted that either E_{0} ⊑_{c} E_{G}^{X} or there is UBMC (see 3.12) reduction of E_{G}^{X} to 2^{<ω_{1}}.

In view of 3.30 and a handful of similar discoveries early in the analysis of equivalence relations, one might hope that understanding turbulence, or any notion of non-classifiability, should ultimately rest on establishing the correct battery of dichotomy theorems. A definition helps to formalize this idea.

Definition 3.31. An orbit equivalent relation is minimum turbulent if it is turbulent and Borel reducible to every other turbulent orbit equivalent. An orbit equivalent relation E_{G}^{X} is minimal turbulent if for all

E ≤_{B} E_{G}^{X}

either:

1. E admits classification by countable models; or

2. E_{G}^{X} ≤_{B} E.

[33] indicated that the structure here is more complicated than the smooth\nonsmooth dichotomy.

Theorem 3.32. There is no minimum turbulent orbit equivalence relation.

The proof of this result required showing ℝ^{ℕ}/l^{1} and ℝ^{ℕ}/c_{0} are ≤_{B}-incomparable, and that the first of these is minimal. So at least we obtain:

Theorem 3.33. ℝ^{ℕ}/l^{1} (equivalently, 2^{ℕ}/I_{2}) is minimal turbulent.^{3}

In view of these developments it seemed natural at the time of the first draft of this manuscript to conjecture that ℝ^{ℕ}/c_{0} is also minimal turbulent. However Mike Oliver in [70] showed that the large mass of ≤_{B} incomparable Borel equivalence relations (all of which are easily seen to be turbulent) from [61] may be reduced to ℝ^{ℕ}/c_{0}, and thus one has:

Theorem 3.34. (Oliver) ℝ^{ℕ}/c_{0} (equivalently, 2^{ℕ}/I_{d}) is not minimal turbulent.^{4}

Of course given 3.33 and the happy glow of Harrington, Kechris, Louveau we might still hope for a simple basis of turbulence. But in 1998 this was refuted by Ilijas Farah:

Theorem 3.35. (Farah; see [16]) There is no finite or even countable collection of turbulent orbit equivalence relation

E_{G_{1}}^{X_{1}}, ...E_{G_{k}}^{X_{k}}, ...

such that for any turbulent orbit equivalence relation E_{G}^{X} there is some k ∈ ℕ with

E_{G_{k}}^{X_{k}} ≤_{B} E_{G}^{X}.

And then the final stake through the heart:

Theorem 3.36. (Farah; see [17]) There is a turbulent orbit equivalence relation which is not ≤_{B} above any minimal turbulent orbit equivalence relation.

In short, we will never be able to produce any kind of dichotomy theorem along the lines of [30] and this avenue may never lead to any insight regarding when an equivalence relation admits classification by countable models.

Instead of continually revising downwards the goals for dichotomy theorems, this manuscript tries to suggest another approach. Rather than attempting to understand various divisions of classifiable versus non-classifiable in terms of a small or finite basis of unruly examples, we may analyze smoothness and classifiability by countable structures by isolating the dynamic properties necessary for their failure. To this end chapter 9 shows that the presence of a turbulent action is a necessary condition for an orbit equivalence relation to deny having at least a UBMC assignment of HC sets as complete invariants, just as the role of a properly generically ergodic action is always implicit in a group action that refuses a UBMC classification by bounded subsets of the countable ordinals.

3As noted in [33], Kechris has shown 2^{ℕ}/I_{2} ≤_{B} ℝ^{ℕ}/l^{1} ≤_{B} 2^{ℕ}/I_{2}.

4Oliver also observes that ℝ^{ℕ}/c_{0} ≤_{B} 2^{ℕ}/I_{d} ≤_{B} ℝ^{ℕ}/c_{0}.

CHAPTER 4 Classifying homeomorphisms

This chapter consists of detailed examples for various kinds of homeomorphism groups.

The classification result at §4.2 for Hom^{+}([0, 1]) is essentially folklore. The nonclassification theorem at §4.3 for the unit square I believe to be new. Ultimately we obtain that the natural notion of isomorphism for homeomorphisms of the unit square is strictly more complicated than the isomorphism relation on countable structures.

First there are some preliminaries.

4.1. Definitions and remarks

Definition 4.1. Hom([0, 1]) is the group of all homeomorphisms of the unit interval, and more generally for K a compact metric space Hom(K) denotes the group of homeomorphisms of K. Hom^{+}([0, 1]) denotes the subgroup of orientation preserving homeomorphisms – that is to say, those π ∈ Hom([0, 1]) with π(0) = 0, π(1) = 1.

We equip Hom(K) with a topology by taking a compatible metric d on K and letting d̂ be defined on Hom(K) by

d̂(π_{1}, π_{2}) = sup{d(π_{1}(x), π_{2}(x)), d(π_{1}^{−1}(x), π_{2}^{−1}(x)) : x ∈ K}.

By compactness of K the topology generated by d̂ does not depend on the choice of d.

Lemma 4.2. For K a compact metric space, Hom(K) is a Polish group.

Proof. First: Multiplication is continuous:- Let U be open with π_{1} ∘ π_{2} ∈ U. Fix ϵ > 0 so that U includes the set of π with d̂(π, π_{1}∘π_{2}) < ϵ. Then by compactness choose positive δ < ϵ/3 such that for all x_{1}, x_{2} ∈ K, d(x_{1}, x_{2}) < δ implies

d(π_{1}(x_{1}), π_{1}(x_{2})) < ϵ/3

and

d(π_{2}^{−1}(x_{1}), π_{2}^{−1}(x_{2})) < ϵ/3.

Then for all σ_{1}, σ_{2} with

d̂(π_{1}, σ_{1}), d̂(π_{2}, σ_{2}) < δ

we have for all x ∈ K

d(π_{2}(x), σ_{2}(x)) < δ
∴ d(π_{1} ∘ π_{2}(x), π_{1} ∘ σ_{2}(x)) < ϵ/3

∴ d(π_{1} ∘ π_{2}(x), σ_{1} ∘ σ_{2}(x)) ≤ d(π_{1} ∘ π_{2}(x), π_{1} ∘ σ_{2}(x)) + d(π_{1} ∘ σ_{2}(x), σ_{1} ∘ σ_{2}(x))

< ϵ/3 + ϵ/3 < ϵ

by the triangle inequality. Similarly we obtain d(π_{2}^{−1} ∘π_{1}^{−1}(x), σ_{2}^{−1} ∘σ_{1}^{−1}(x)) < 2ϵ/3 all x and so d̂(π_{1} ∘ π_{2}, σ_{1} ∘ σ_{2}) ≤ 2ϵ/3 < ϵ.

It is immediate from the definitions that the function π ↦ π^{−1} is continuous on Hom(K).

We obtain completeness of the metric by adapting the usual proof that a uniformly convergent sequence of functions has a continuous limit. Suppose (π_{i})_{i∈ℕ} is d̂-Cauchy. By definition of the metric we obtain for all x that (π_{i}(x))_{i∈ℕ} and (π_{i}^{−1}(x))_{i∈ℕ} are d-Cauchy. So we may define π_{∞}, σ_{∞} : K → K by

π_{∞}(x) = lim_{i∈ℕ}π_{i}(x),
σ_{∞}(x) = lim_{i∈ℕ}π_{i}^{−1}(x).

These are both continuous and we just need that they pair as each’s inverse for π_{∞} ∈ Hom(K). As i → ∞ we have firstly that

π_{i}(x) → π_{∞}(x)

and secondly, by the uniformity of the convergence, that

σ_{∞}(π_{i}(x)) → π_{i}^{−1}(π_{i}(x)) = x.

Thus σ_{∞}(π_{∞}(x)) = x all x ∈ K and hence σ_{∞} = (π_{∞})^{−1}.

Finally. Separability:- For ϵ > 0 and x ∈ K let B_{ϵ}(x) = {y ∈ K : d(x, y) < ϵ}. For 0 < ϵ_{0} < ϵ_{1} let H_{ϵ_{0},ϵ_{1}} be the set of π ∈ Hom(K) for which all x_{1}, x_{2} in K with d(x_{1}, x_{2}) < ϵ_{0} has

d(π(x_{1}), π(x_{2})), d(π^{−1}(x_{1}), π^{−1}(x_{2})) < ϵ_{1}.

Compactness implies for all ϵ_{1} and π there is some ϵ_{0} with π ∈ H_{ϵ_{0},ϵ_{1}}. We only need show for 0 < ϵ_{0} < ϵ_{1} < 2ϵ_{1} < ϵ_{2} that H_{ϵ_{0},ϵ_{1}} has a countable subset that is within d̂-distance ϵ_{2} of every other element of H_{ϵ_{0},ϵ_{1}}. But if k_{0}, k_{1}, ..., k_{n} ∈ K is so that ⋃ B_{ϵ_{0}}(k_{i}) covers K and if {π_{i} : i ∈ ℕ} is chosen so that

{(π_{i}(k_{0}), π_{i}(k_{1}), ..., π_{i}(k_{n}), π_{i}^{−1}(k_{0}), ..., π_{i}^{−1}(k_{n})) : i ∈ ℕ}

is dense (as a subset of K^{2n+2}) in

{(π(k_{0}), π(k_{1}), ..., π^{−1}(k_{n})) : π ∈ H_{ϵ_{0},ϵ_{1}}}

then for any π ∈ H_{ϵ_{0},ϵ_{1}} we may choose π_{i} so for all j

d(π_{i}(k_{j}), π(k_{j})), d(π_{i}^{−1}(k_{j}), π^{−1}(k_{j})) < ϵ_{2} − 2ϵ_{1}.

Then for all k ∈ K we may find some k_{j} with d(k, k_{j}) < ϵ_{0}, and conclude that

d(π_{i}(k), π(k)) ≤ d(π_{i}(k), π_{i}(k_{j})) + d(π_{i}(k_{j}), π(k_{j})) + d(π(k_{j}), π(k))
< ϵ_{1} + (ϵ_{2} − 2ϵ_{1}) + ϵ_{1} < ϵ_{2}

and similarly

d(π_{i}^{−1}(k), π(k)^{−1}) < ϵ_{1} + (ϵ_{2} − 2ϵ_{1}) + ϵ_{1} < ϵ_{2}

as needed.

Exercise 4.3. (a) In the case of K = [0, 1] and {k_{i} : i ∈ ℕ} ⊂ [0, 1] dense we may take as a basis for Hom([0, 1]) sets of the form

{σ : |σ(k_{i}) − π(k_{i})| < ϵ for i = 0, 1, ..., n}

as n ranges over ℕ, π over Hom(K), ϵ over the positive reals.

(b) For K = [0, 1]^{2} this pointwise topology is already weaker the one than generated by the d̂ of 4.1.

(c) For K compact we obtain as a subbasis for the topology of Hom(K) sets of the form

{π : π[C] ⊂ O}

for O ⊂ K open and C ⊂ K closed.

Exercise 4.4. Show that every element of Hom^{+}([0, 1]) is torsion free, and thus S_{∞} is not homeomorphically isomorphic to a closed subgroup of Hom^{+}([0, 1]).

4.2. Classification in dimension 1

Notation 4.5. ≈ denotes the conjugacy equivalence relation on Hom^{+}([0, 1])

– so that for π_{1}, π_{2} ∈ Hom^{+}([0, 1]) we set π_{1} ≈ π_{2} if there is some σ ∈ Hom^{+}([0, 1]) with

σ ∘ π_{1} ∘ σ^{−1} = π_{2}.

Let ≈′ denote conjugacy on the full Hom([0, 1]) and ≈′′ on Hom([0, 1]^{2}).

Theorem 4.6. (folklore) There is a countable language L such that

≈≤_{B}≅|_{Mod(L)}.

Proof. Let L be generated by the symbols <, P_{+}, P_{−}, P_{=}, with < a binary relation, P_{+}, P_{−}, P_{=} unary predicates. For π ∈ Hom^{+}([0, 1]) define the underlying set of M(π) to consist of the disjoint union of (P_{+})^{M(π)}, (P_{−})^{M(π)}, (P_{=})^{M(π)}, where:

(i) (P_{+})^{M(π)} is the set of all open intervals (x, y) such that x is less than y, both are in the unit interval, π(x) = x and π(y) = y, and for all z ∈ (x, y) one has π(z) > z; (ii) (P_{−})^{M(π)} is the set all open intervals (x, y) such that x is less than y, both are in the unit interval, π(x) = x and π(y) = y, and for all z ∈ (x, y) one has π(z) < z; (iii) (P_{=})^{M(π)} is the set of all open intervals (x, y) in [0, 1] that are maximal with respect to the property of π|_{(x,y)} = id|_{(x,y)} – in other words

∀z ∈ (x, y)(π(z) = z),

but for all ϵ > 0 there exist z_{0} ∈ (x − ϵ, x) and z_{1} ∈ (y, y + ϵ)

π(z_{0}) ≠ z_{0},
π(z_{1}) ≠ z_{1}.

For (x, y), (x′, y′) ∈ M(π) we set (x, y) < (x′, y′) if and only if y ≤ x′. Since the intervals are non-overlapping this defines a strict linear ordering on the countable structure M(π). It is more or less immediate that π_{1} ≈ π_{2} ⇒ M(π_{1}) ≅ M(π_{2}). Strictly speaking we need to build a structure with ℕ as its underlying set, but there are no special difficulties in obtaining in a Borel fashion from π a model isomorphic to M(π) in Mod(L). So this leaves us just with showing that

M(π_{1}) ≅ M(π_{2}) ⇒ π_{1} ≈ π_{2}.

So fix π_{1}, π_{2} ∈ Hom^{+}([0, 1]), and Φ : M(π_{1}) ≅ M(π_{2}) an isomorphism of models.

Claim: For all (x, y) ∈ M(π_{1}) there exists an orientation preserving homeomorphism

σ_{x} : (x, y) → Φ((x, y))
π_{1}|_{(x,y)} = σ_{x}^{−1} ∘ π_{2}|_{Φ((x,y))} ∘ σ_{x}.

Proof of claim: Assume that (x, y) ∈ (P_{+})^{M(π_{1})}; the other case of (x, y) ∈ (P_{−})^{M(π_{1})} is exactly similar and that of (x, y) ∈ (P_{=})^{M(π_{1})} trivial. π_{1} is strictly increasing on (x, y), so π_{2} is strictly increasing on Φ((x, y)) =_{df} (x′, y′) by the assumption on Φ.

Choose any z_{0} ∈ (x, y) and w_{0} ∈ (x′, y′) and let z_{i} = (π_{1})^{i}(z_{0}), w_{i} = (π_{2})^{i}(w_{0}) for i ∈ ℤ. Fix some order preserving homeomorphism σ′ : [z_{0}, z_{1}] → [w_{0}, w_{1}]. Now note that there will be a unique extension σ_{x} : (x, y) → (x′, y′) with σ_{x}((π_{1})^{i}(z)) = (π_{2})^{i}(σ′(z)) for any z ∈ [z_{0}, z_{1}]. (□Claim)

Now take σ′′ = ⋃{σ_{x} : (x, y) ∈ M(π_{1})}. σ′′ is defined on an open dense subset of [0, 1], has open dense range, is order preserving, and hence σ′′ extends to some σ ∈ Hom^{+}([0, 1]). Since for a dense set of x ∈ [0, 1]

σ^{−1}(π_{2}(σ(x))) = π_{1}(x)

it follows that that σ conjugates π_{1} and π_{2}.

Lemma 4.7. Let G be a Polish group and H < G a clopen subgroup. Let X be a Polish G-space. If E_{H}^{X} admits classification by countable models, so does E_{G}^{X}.

Proof. Let θ : X → Mod(L) be a Borel function witnessing the assumption on E_{H}^{X}, and F a fresh binary relation. Take L′ to be the language generated by L and F. Since G is separable and H open we can find {Hg_{i} : i ∈ ℕ} a countable enumeration of the left cosets of H. We will define θ′(x) ∈ Mod(L′) to be something like the disjoint union of {θ(g_{i} · x) : i ∈ ℕ}.

Now for any x ∈ X we can take θ′(x) to be an L′ model such that: F defines an equivalence relation on θ′(x); for (A_{i})_{i∈ℕ} some enumeration of the F -equivalence classes we have that each A_{i} is isomorphic as an L-structure to θ(g_{i} · x); for S a relation symbol in L and a⃗ ∈ A_{i}^{<ℕ}, we have

(θ(x) ⊧ S(a⃗, b⃗)) ⇒ b⃗ ∈ A_{i}^{<ℕ}.

Lemma 4.8. Let ∼′ denote conjugacy on Hom([0, 1]) by elements in Hom^{+}([0, 1]), so that π_{1}, π_{2} ∈ Hom([0, 1]) are ∼′-equivalent if there is some σ ∈ Hom^{+}([0, 1]) so that

σπ_{1}σ^{−1} = π_{2}.

Then ∼′ admits classification by countable models.

Proof. In light of 4.6 it suffices to assign countable models to π : [0, 2] → [0, 2] that are orientation reversing and fixing 1 – that is,

π(0) = 2, π(1) = 1, π(2) = 0.

(The transition here from Hom([0, 1]) to Hom([0, 2]) is clearly harmless, and any orientation reversing map must have a fixed point, which we may as well assume to be 1.) For such π : [0, 2] → [0, 2] orientation reversing we let N (π) = M(π^{2}|_{[0,1]}), where M(π^{2}|_{[0,1]}) is chosen according to 4.6. It suffices to show that if N (π_{1}) ≅ N (π_{2}) then π_{1} ∼′ π_{2}.

So fix π_{1}, π_{2} with N (π_{1}) ≅ N (π_{2}). By 4.6 find σ_{0} ∈ Hom^{+}([0, 1]) with

σ_{0} ∘ (π_{1})^{2}|_{[0,1]} ∘ σ_{0}^{−1} = (π_{2})^{2}|_{[0,1]}.

Let π̂_{1} = σ̂_{0}π_{1}σ̂_{0}^{−1}, where

σ̂_{0}|_{[0,1]} = σ_{0},

σ̂_{0}|_{[1,2]} = id|_{[1,2]}.

It suffices to show π̂_{1} and π_{2} are conjugated by an element of Hom^{+}([0, 2]). Choose σ_{1} ∈ Hom^{+}([1, 2]) such that σ_{1} ∘ π̂_{1}|_{[0,1]} = π_{2}|_{[0,1]}; let

σ̂_{1} ∈ Hom^{+}([0, 2]),
σ̂_{1}|_{[0,1]} = id_{[0,1]},
σ̂_{1}|_{[1,2]} = σ_{1}.

At once

σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1}|_{[0,1]} = π_{2}|_{[0,1]};

so to obtain

σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1}|_{[1,2]} = π_{2}|_{[1,2]}

it suffices to check that

(σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1})|_{[1,2]} ∘ (σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1})|_{[0,1]} = (σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1})^{2}|_{[0,1]}

is equal to

(π_{2})^{2}|_{[0,1]} = π_{2}|_{[1,2]} ∘ π_{2}|_{[0,1]} = π_{2}|_{[1,2]} ∘ σ̂_{1} ∘ π̂_{1} ∘ σ̂_{1}^{−1}|_{[0,1]}.

Since σ̂_{1}|_{[0,1]} = id|_{[0,1]} this would amount to asserting

(π̂_{1})^{2}|_{[0,1]} = (π_{2})^{2}|_{[0,1]}

which is exactly

(σ̂_{0} ∘ π_{1} ∘ σ̂_{0}^{−1})^{2}|_{[0,1]} = (π_{2})^{2}|_{[0,1]}.

Theorem 4.9. The orbit equivalence relation ≈′ induced by the self action of Hom([0, 1]) by conjugation admits classification by countable models.

Proof. By 4.8 and 4.7 since Hom^{+}([0, 1]) clopen in Hom([0, 1]).

On the other hand we obtain a kind of converse using the following from [20]:

Theorem 4.10. (Friedman, Stanley) Let < be a binary relation, and define the space Mod(LO) be the class of {<}-structures on ℕ that are linear orderings. Then for any countable language L

≅|_{Mod(L)} ≤_{B}≅|_{Mod(LO)}.

Proof. First assume that L is generated by a single binary relation R. Fixing an L-structure M we describe a countable linear ordering L(M). The linear ordering will not actually have ℕ as its underlying set, so we leave to the reader the routine verifications that this may be reorganized to provide in a Borel fashion a linear ordering of ℕ isomorphic to L(M).

Let η be some transcendental number and let

s(·, ·) : ℕ^{2} ↪ ℕ

be a pairing function. Let (p_{i})_{i∈ℕ} enumerate the primes greater than 3. For each a⃗ = ⟨a_{0}, a_{1}, ..., a_{n}⟩ in M^{<ℕ} we associate the natural number

N(a⃗) = 3^{n} · ∏_{i,j≤n} p_{s(i,j)}^{ϵ_{i,j}},

where ϵ_{i,j} = 3 if a_{i} ≠ a_{j} and M ⊧ R(a_{i}, a_{j}), ϵ_{i,j} = 2 if a_{i} ≠ a_{j} and M ⊧ ¬R(a_{i}, a_{j}), ϵ_{i,j} = 1 if a_{i} = a_{j} and M ⊧ R(a_{i}, a_{j}), ϵ_{i,j} = 0 if a_{i} = a_{j} and M ⊧ ¬R(a_{i}, a_{j}).

Thus N(a⃗) is a natural number that encodes the length of a⃗ and how the various elements of the sequence are related.

We let B_{i} = (1, 2) ∩ (ℚ + (η^{i})). Let

B = ⋃_{i∈ℕ} B_{i}.

The relevant features are that the B_{i}’s partition B into countably many dense subsets, B is a linear ordering without endpoints, B is dense in itself (in the sense that any open interval in B contains some point), and that any non-positive integer is smaller than any element in B.

We then let L(M) consist of all finite sequences of the form ⟨b_{0}, ..., b_{n−1}, −k⟩ where

(i) each b_{i} ∈ B;

(ii) k ∈ ℕ, and k < N(a⃗) where ⟨a_{0}, ..., a_{n−1}⟩ is defined by having at each

j
b_{j} ∈ B_{a_{j}}.

We give L(M) the lexicographic ordering. Thus

⟨b_{0}, ..., b_{n−1}, −k⟩ < ⟨b′_{0}, ..., b′_{m−1}, −k′⟩

if either (i) there is i < min(n, m) with b_{i} ≠ b′_{i}, and for i least such we have b_{i} < b′_{i}; or (ii) n < m and for i < n we have b_{i} = b′_{i}; or (iii) ⟨b_{0}, ..., b_{n−1}⟩ = ⟨b′_{0}, ..., b′_{m−1}⟩ and −k < −k′.

As a rough picture, the idea of this construction is that we begin with a linear ordering consisting of copies of finite sequences from M, so that each sequence appears in some sense ℚ-often, and to the near right of any appearance of a sequence we have appearances of all its possible extensions. If at each i ≤ n we have some b_{i} ∈ B_{a(i)} for some a(i) ∈ M, we should choose to think of the sequence ⟨b_{0}, ..., b_{n}⟩ as representing the sequence ⟨a(0), a(1), ..., a(n)⟩. Since each B_{i} is dense, there are not only many representatives of each ⟨a(0), a(1), ..., a(n)⟩, but moreover given ⟨b_{0}, ..., b_{n}⟩ and a ∈ M there are many choices of b ∈ B_{a} with which we can extend our sequence to ⟨b_{0}, ..., b_{n}, b⟩ so that it now represents ⟨a(0), a(1), ..., a(n), a⟩. We have then taken the further step of replacing each point of the form ⟨b_{0}, ..., b_{n}⟩ with a finite discrete linear ordering whose size indicates how the terms in the corresponding ⟨a(0), a(1), ..., a(n)⟩ ∈ M^{<ℕ} sequence are related.

Although the ideas are similar in both cases, the easier direction of the proof of the theorem is in showing M ≅ N implies L(M) and L(N) are isomorphic. So let us concentrate instead on proving the converse and suppose that

π : L(M) ≅ L(N)

is an isomorphism between the linear orderings. Let E_{π} be the set of all pairs of sequences

(a⃗, c⃗) = (⟨a_{0}, ..., a_{m}⟩, ⟨c_{0}, ..., c_{n}⟩) ∈ M^{<ℕ} × N^{<ℕ}

such that

⟨b_{0}, ..., b_{m}, −k⟩ ∈ L(M)

and

⟨b′_{0}, ..., b′_{m}, −k′⟩ ∈ L(N)

with each b_{i} ∈ B_{a_{i}}, each b′_{i} ∈ B_{c_{i}} and

π(⟨b_{0}, ..., b_{m}⟩) = ⟨b′_{0}, ..., b′_{n}⟩.

Claim(I) (⟨a_{0}, ..., a_{m}⟩, ⟨c_{0}, ..., c_{n}⟩) ∈ E_{π} implies that

(a) m = n; and for all i, j ≤ m
(b) M ⊧ R(a_{i}, a_{j}) if and only if N ⊧ R(c_{i}, c_{j});
(c) a_{i} = a_{j} if and only c_{i} = c_{j}.

Proof of claim. Note that N(a⃗) is equal to the size of the maximal finite interval containing ⟨b_{0}, ..., b_{m}, −k⟩ in L(M). (That is to say: There are x, x̂ ∈ L(M) with x ≤ ⟨b_{0}, ..., b_{m}, −k⟩ ≤ x̂ having exactly N(a⃗) many objects in the closed interval [x, x̂]^{L(M)}, and moreover for any y < x and ŷ > x̂, the intervals [y, x]^{L(M)} and [x̂, ŷ]^{L(M)} are both infinite.) Since the same holds on the N side and π is an isomorphism of linear orderings, it follows that N(a⃗) = N(c⃗), and thus we have the claim by the definition of N(a⃗). (Claim□)

Claim(II) For all (⟨a_{0}, ..., a_{m}⟩, ⟨c_{0}, ..., c_{m}⟩) ∈ E_{π} and a ∈ M there is some c ∈ N with

(⟨a_{0}, ..., a_{m}, a⟩, ⟨c_{0}, ..., c_{m}, c⟩) ∈ E_{π}.

Proof of claim. Fix ⟨b_{0}, ..., b_{m}, −k⟩ ∈ L(M) and ⟨b′_{0}, ..., b′_{m}, −k′⟩ ∈ L(N) witnessing the definition of (⟨a_{0}, ..., a_{m}⟩, ⟨c_{0}, ..., c_{m}⟩) ∈ E_{π} – that is to say, each b_{i} ∈ B_{a_{i}}, b′_{i} ∈ B_{c_{i}}, and π(⟨b_{0}, ..., b_{m}, −k⟩) = ⟨b′_{0}, ..., b′_{m}, −k′⟩. Choose b ∈ B_{a} and let

π(⟨b_{0}, ..., b_{m}, b, 0⟩) = ⟨b̄′_{0}, ..., b̄′_{m}, b̄_{m+1}, −k̄⟩.

Note that there is no maximal finite closed interval strictly between ⟨b_{0}, ..., b_{m}, −k⟩ and ⟨b_{0}, ..., b_{m}, b, 0⟩ whose size is indivisible by 3^{m+1}. Therefore since π is an isomorphism there can be no maximal finite closed interval strictly between ⟨b′_{0}, ..., b′_{m}, −k′⟩ and ⟨b̄_{0}, ..., b̄′_{m}, b̄_{m+1}, −k̄⟩ whose size is indivisible by 3^{m+1}. Considering the definition of the linear orderings we obtain

⟨b̄′_{0}, ..., b̄′_{m}, b̄_{m+1}, −k̄⟩ = ⟨b′_{0}, ..., b′_{m}, b′, −k̄⟩,

some b′, and then for c such that b′ ∈ B_{c} we have ⟨c_{0}, ..., c_{m}, c⟩ as required. (Claim□)

Claim(III) For all (⟨a_{0}, ..., a_{m}⟩, ⟨c_{0}, ..., c_{m}⟩) ∈ E_{π} and c ∈ N there is some a ∈ M with

(⟨a_{0}, ..., a_{m}, a⟩, ⟨c_{0}, ..., c_{m}, c⟩) ∈ E_{π}.

Proof of claim. As in (II). (Claim□) We can now use a technique known in model theory as a back-and-forth argu-

ment, and dating in some sense to the work of Cantor, to obtain an isomorphism between M and N.

We first let (x_{i})_{i∈ℕ} enumerate M and (y_{i})_{i∈ℕ} enumerate N. We then build sequences of a_{0}, a_{1}, ..., a_{i}, ... ∈ M and c_{0}, c_{1}, ..., c_{i}, ... ∈ N such that

(i) each (⟨a_{0}, ...a_{i}⟩, ⟨c_{0}, ..., c_{i}⟩) ∈ E_{π};

(ii) each x_{n} ∈ {a_{0}, ...a_{2n}};
(iii) each y_{n} ∈ {c_{0}, ..., c_{2n+1}}.

We build this sequence inductively, using claim(II) to propagate (ii), claim(III) for (iii). We then define

ρ : M → N

by

a_{i} ↦ c_{i}.

By claim(I)(c) this is well defined. By (iii) it is onto and by (ii) it is everywhere defined. Claim(I)(b) requires it to respect the relation R, and therefore provide an isomorphism between M and N.

We skipped over the easier direction of the equivalence

M ≅ N ⇔ L(M) ≅ L(N),

but it resembles the back and forth argument above. The main point is that given any bijection π of ℕ, we may find an isomorphism

ρ : (B, <) ≅ (B, <)

such that at each i

ρ[B_{i}] = B_{π(i)}.

We are only left to describe the general construction for general L. For L a countable relational language language, we let (L_{n}) be an increasing sequence of finite languages with

L = ⋃_{n∈ℕ} L_{n}.

We then proceed as before, but with the one change that N(⟨a_{0}, a_{1}, ..., a_{n}⟩) will be a natural number that encodes for us how the various elements of the sequence ⟨a_{0}, a_{1}, ..., a_{n}⟩ are related over L_{n}.

Corollary 4.11. (folklore) Let L be a countable language. Then

≅|_{Mod(L)} ≤_{B}≈ .

Proof. Following the proof of 4.6 we may encode any countable linear ordering by an element of Hom^{+}([0, 1]) considered up to conjugacy. Now it follows by 4.10.

Exercise 4.12. Let M ∈ Mod(LO). Then the automorphism group of M, Aut(M), can be realized as a closed subgroup of Hom^{+}([0, 1]).

Exercise 4.13. The shift action of Hom^{+}([0, 1]) on F([0, 1]), given by

π · C = {π(x) : x ∈ C},

admits classification by countable models.

Exercise 4.14. For L a countable language, ≅|_{Mod(L)} ≤_{B}≈′ .

4.3. Non-classification in dimension 2

Notation 4.15. For the remainder of this section let G = ∏_{ℕ} ℤ, with the product group and topological structure, under which it becomes a Polish group. Define

Ĝ = {x⃗ ∈ G : x_{n}/n → 0};

so in some sense this is the subgroup consisting of sequences which do not stray too quickly from 0. Note that Ĝ is a Polishable group, as witnessed by the complete metric

d(x⃗, y⃗) = sup_{ℕ}|(x_{n} − y_{n})/n|.

Lemma 4.16. The coset equivalence relation of Ĝ on G does not admit classification by countable structures.

Proof. The action of Ĝ on G is turbulent: Clearly every orbit is dense; each orbit is meager by 2.56 since Ĝ = Ĝ^{−1}Ĝ contains no open sets; as for condition (iii) in the definition of turbulence, note that if g⃗ ∈ G,

U ⊃ {h⃗ ∈ G : h⃗|_{n} = g⃗|_{n}}

and

V ⊃ {h⃗ ∈ Ĝ : d(h⃗,0) < 1/n},

(where 0 = (0, 0, ...) is the identity in Ĝ) then

\overline{O(h⃗, U, V)} ⊃ \{g⃗ ∈ G : g⃗|_{n+1} = h⃗|_{n+1}\}.

Theorem 4.17. The conjugacy equivalence relation ≈′′ on Hom([0, 1])^{2} does not admit classification by countable models.

Proof. It suffices to show that

EG ; Ĝ ≤_{B}≈′′ .

The notation is smoother working with Hom([−1, 1] × [0, 1]). We define closed rectangles I_{n,k} inside [0, 1] × [0, 1] for n ∈ ℕ, n > 0, k ∈ ℤ, and for that purpose fix an order preserving homeomorphism f : ℝ → (0, 1). The closed rectangle I_{n,k} has vertices

(1/2n, f(k/n)), (1/(2n + 1), f(k/n))

and (1/2n, f((k + 1)/n)), (1/(2n + 1), f((k + 1)/n)).

Also define closed rectangles J_{0}, J_{1}, with J_{0} occupying the top half of [−1, 0]×[0, 1] and having vertices

(−1, 1), (0, 1),

and (−1, 1/2), (0, 1/2),

and J_{1} occupying the bottom half with vertices

(−1, 1/2), (0, 1/2),

and (−1, 0), (0, 0).

We then choose π_{n,k} ∈ Hom(I_{n,k}), for all n, k, π_{0} ∈ Hom(J_{0}), and π_{1} ∈ Hom(J_{1}) such that:

(i) for all (x, y) in (I_{n,k})^{o} (the interior of I_{n,k}), π_{n,k}(x, y) = (x, y′) for some y′ > y; (ii) as (x, y) approaches ∂(I_{n,k}) (the boundary of I_{n,k}), the distance d(π_{n,k}(x, y), (x, y)) (the usual euclidean distance from π_{n,k}(x, y) to (x, y)) approaches 0; (iii) as (x, y) → [−1, 0] × [0, 1]

d(π_{n,k}(x, y), (x, y)) → 0

uniformly in n, k, in the sense that for all δ > 0 there will be some δ′ > 0 for whom all n, k and (x, y) ∈ I_{n,k} have

d((x, y), [−1, 0] × [0, 1]) < δ′ ⇒ d(π_{n,k}(x, y), (x, y)) < δ;

(iv) as (x, y) in J_{0} approaches ∂(J_{0}) (the boundary of J_{0} as evaluated in ℝ^{2})

d(π_{0}(x, y), (x, y)) → 0;

J_{0} J_{1}  ^{I_{1,0}}^{I_{1,1}}^{I_{1,2}}^{I_{1,−1}}^{I_{1,−2}}^{I_{2,0}}^{I_{2,1}}^{I_{2,2}}^{I_{2,3}}

Figure 4.1. The homeomorphisms defined region by region

(v) as (x, y) in J_{1} approaches ∂(J_{1})

d(π_{1}(x, y), (x, y)) → 0;

(vi) for all (x, y) ∈ (J_{0})^{o} we have π_{0}(x, y) = (x′, y) some x′ > x; (vii) for all (x, y) ∈ (J_{1})^{o} we have π_{0}(x, y) = (x′, y) some x′ < x.

We also need to set σ_{n,k,l} : I_{n,k} → I_{n,k+l} given by

(x, f(y)) ↦ (x, f(y + l/n)),

and assume that this is chosen previously in harmony with the π_{n,k} so that

π_{n,k+l} = σ_{n,k,l} ∘ π_{n,k} ∘ σ_{n,k,l}^{−1}.

We further need that the orbits of any point flow as far as possible granting the above restrictions. So:

(viii) for all (x, y), (x, y′) ∈ (I_{n,k})^{o} there exists i ∈ ℤ and y′′ > y′ such that

(π_{n,k})^{i}(x, y) = (x, y′′);

(ix) for all (x, y), (x′, y) ∈ (J_{j})^{o} (j = 0, 1) there exists i ∈ ℤ and x′′ > x′ such that

(π_{j})^{i}(x, y) = (x′′, y).

Assuming all the above define

ρ : G → Hom([−1, 1] × [0, 1]).

For g⃗ ∈ G define ρ(g⃗) by the requirement that:

(a) ρ(g⃗)|_{[−1,0]×[0,1]} = π_{0} ∪ π_{1}; (bi) for g⃗(n) = k

ρ(g⃗)|_{I_{n,k}} = π_{n,k}^{−1};

(bii) for g⃗(n) ≠ k

ρ(g⃗)|_{I_{n,k}} = π_{n,k};

(c) at all other (x, y) ∈ [−1, 1]×[0, 1] we have simply ρ(g⃗)(x, y) = (x, y).

I_{n,g(n)} I_{n,g(n)−1} I_{n,g(n)−2} I_{n,g(n)+1} I_{n,g(n)+2}

Figure 4.2. A closer look

Claim(I): For all g⃗, h⃗ ∈ G if π ∈ Hom([−1, 1] × [0, 1]) with

π ∘ ρ(g⃗) ∘ π^{−1} = ρ(h⃗)

then

π[[−1, 0] × [0, 1]] = [−1, 0] × [0, 1],
π[{0} × [0, 1]] = {0} × [0, 1],
π[{0} × [0, 1/2]] = {0} × [0, 1/2],

and so the point (0, 1/2) is fixed by π.

Proof of claim: Let D_{0}(h⃗) be the set of all z ∈ ([−1, 1] × [0, 1])^{o} for which there is some z′ ∈ ∂([−1, 1] × [0, 1]) with

(ρ(h⃗))^{i}(z) → z′

as

i → −∞.

Define D_{1}(h⃗) to be the z ∈ ([−1, 1]×[0, 1])^{o} for which there is some z′ ∈ ∂([−1, 1]× [0, 1]) with

(ρ(h⃗))^{i}(z) → z′

as

i → +∞.

D_{0}(h⃗) = (J_{0})^{o} and D_{1}(h⃗) = (J_{1})^{o} and

{0} × (1/2, 1) = ∂(D_{0}(h⃗)) ∩ ([−1, 1] × [0, 1])^{o}

and

{0} × (0, 1/2) = ∂(D_{1}(h⃗)) ∩ ([−1, 1] × [0, 1])^{o}.

Similarly

{0} × [1/2, 1) = ∂(D_{0}(g⃗)) ∩ ([−1, 1] × [0, 1])^{o}

and

{0} × (0, 1/2] = ∂(D_{1}(g⃗)) ∩ ([−1, 1] × [0, 1])^{o}.

By the nature of the definitions π[D_{0}(g⃗)] = D_{0}(h⃗), π[D_{1}(g⃗)] = D_{1}(h⃗), thereby giving the claim. (Claim(I)□)

Claim(II) If g⃗, h⃗ ∈ G and π ∈ Hom([−1, 1] × [0, 1]) with

π ∘ ρ(g⃗) ∘ π^{−1} = ρ(h⃗)

then for all n ∈ ℕ

π[I_{n,g(n)}] = I_{n,h(n)}.

Proof of claim. From (I), [0, 1] × [0, 1] is a π invariant set. Since the strips of the form

(\frac{1}{n + 1}, \frac{1}{n}) × [0, 1]

are seen to be invariant by a mathematical induction on n, and then since each (I_{n,m})^{o} is a maximal connected open set in [0, 1]^{2} on which ρ(g⃗) and ρ(h⃗) have no fixed points, we obtain π(I_{n,g(n)}) = I_{n,m} some m. We can characterize (I_{n,g(n)})^{o} inside {(I_{n,k})^{o} : k ∈ ℤ} as unique (I_{n,k})^{o} for which there is some other

(I_{n,l})^{o} ≠ (I_{n,k})^{o}

with

{lim_{i→+∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,l})^{o}} =
{lim_{i→+∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,k})^{o}}.

The same definition works for ρ(h⃗) to describe h(n), and so we have m = h(n). (□Claim(II))

Claim(III) If g⃗, h⃗ ∈ G and π ∈ Hom([−1, 1] × [0, 1]) with

π ∘ ρ(g⃗) ∘ π^{−1} = ρ(h⃗)

then for all n ∈ ℕ and l ∈ ℤ

π[I_{n,g(n)+l}] = I_{n,h(n)+l}.

Proof of Claim: We can specify each (I_{n,h(n)+1})^{o} inside {(I_{n,k})^{o} : k ∈ ℤ} as the unique (I_{n,k})^{o} for which

{lim_{i→−∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,g(n)})^{o}} =
{lim_{i→−∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,k})^{o}}.

Inductively for l > 0 we may describe (I_{n,g(n)+l+1})^{o} as the unique (I_{n,k}) with

{lim_{i→−∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,k})^{o}} =
{lim_{i→+∞}ρ(g⃗)^{i}(z) : z ∈ (I_{n,g(n)+l})^{o}}.

A similar characterization holds for the (I_{n,g(n)−l})^{o}’s, l > 0, and all these formulations work symmetrically on the h⃗ side. (□Claim(III))

Then for g⃗, h⃗, π as above, claims (I) and (III) above imply that

d(I_{n,0}, I_{n,h(n)−g(n)}) → 0

n → 0. Thus

|f(\frac{h(n) − g(n)}{n}) − f(0)| → 0

and hence

\frac{h(n) − g(n)}{n} → 0

which amounts to asserting that h⃗ − g⃗ ∈ Ĝ.

Conversely, if h⃗ − g⃗ ∈ Ĝ then we may define σ_{0} ∈ Hom([0, 1]^{2}) so that

σ_{0}|_{I_{n,l}} = σ_{n,l,h(n)−g(n)};

on the strips of the form

(\frac{1}{2n + 2}, \frac{1}{2n + 1}) × [0, 1],

for n > 0, take the “average” of

σ_{0}|_{(1/(2n+1),1/(2n))×[0,1]}

and

σ_{0}|_{(1/(2n+3),1/(2n+2))×[0,1]},

so that for z ∈ ℝ, λ ∈ (0, 1) let σ_{0}(\frac{1}{2n+1+λ}, f(z)) equal

(\frac{1}{2n + 1 + λ}, f(z + \frac{(h(n) − g(n))(1 − λ)}{n} + \frac{(h(n + 1) − g(n + 1))λ}{n + 1})).

On the strip (1/2, 1) × [0, 1] average σ_{0}|_{(1/4,1/2)×[0,1]} and the identity, so that

σ_{0}(\frac{1}{1 + λ}, f(z)) = (\frac{1}{1 + λ}, f(z + \frac{h(1) − g(1)}{λ})).

By the assumption that |g(n) − h(n)|/n → 0 as n → ∞

d(z, σ_{0}(z)) → 0

as z → ∂([0, 1]^{2}). Hence in particular σ_{0} extends to σ ∈ Hom([−1, 1] × [0, 1]) with σ|_{[−1,0]×[0,1]} equal to the identity. It follows at once from the definitions of the π_{n,k} and σ_{n,k,l} that σ conjugates ρ(g⃗) and ρ(h⃗).

Su Gao had previously noted that c_{0} is a closed subgroup of Hom([0, 1]^{2}); the above argument was inspired by his construction.

Theorem 4.18. ≅|_{Mod(L)} <_{B}≈′′.

Proof. Given the preceding theorem and 4.11 it suffices to show that ≈≤_{B}≈′′ which in turn follows various ways.

For instance comparing the proof of 4.17 we may associate to π ∈ Hom^{+}([0, 1]) a homeomorphism σ(π) ∈ Hom([−1, 2] × [0, 1]) so that, for f : ℝ → (0, 1) a homeomorphism as before,

σ(π)(−1 + λ, f(y)) = (−1 + λ, f(y + 2λ)),
σ(π)(−λ, f(y)) = (−λ, f(y + 2λ))

for λ ∈ [0, 1/2], and

σ(π)|_{[1,2]×[0,1]} = id|_{[1,2]×[0,1]}.

Thus this is a homeomorphism that shunts every point in (−1, 0)×(0, 1) up towards the top line (−1, 0)×{1} and uniformly fades away to the identity at the boundary. On [0, 1] × [0, 1] we have

(σ(π))(x, y) = (π(x), y).

Hence (σ(π))|_{[0,1]×{0}} is conjugate by a homeomorphism to π.

Note that the boundary ∂([−1, 2] × [0, 1]) is definable in terms of properties which are internal to [−1, 2] × [0, 1] viewed purely as a topological space. (−1, 0) × {1} is characterized as the maximal (non-trivial) line segment included in the intersection of

∂([−1, 2] × [0, 1])

and

{lim_{i∈ℕ}(σ(π))^{i}(z) : z ∈ [−1, 2] × [0, 1], (σ(π))(z) ≠ z}.

We then can characterize (−1, 0) × (0, 1) as the points whose limit under σ(π) tend towards the set (−1, 0) × {1}.

Thus if π_{1}, π_{2} ∈ Hom^{+}([0, 1]), τ ∈ Hom([−1, 2] × [0, 1]),

σ(π_{1}) = τ^{−1} ∘ σ(π_{2}) ∘ τ,

then (−1, 0)×(0, 1) must be τ invariant. One can then obtain that (σ(π_{1}))|_{[0,1]×{0}}, (σ(π_{2}))|_{[0,1]×{0}} are conjugate by an orientation preserving homeomorphism, and thus π_{1}, π_{2} are conjugate.

The lengthy but ultimately routine verification that

π ↦ σ(π)

is Borel can be left as an exercise.

Definition 4.19. Let X(M) be the set of infinite double nested sequences (x_{i,j})_{i,j∈ℕ} such that

(i) each x_{i,j} ∈ ℝ^{≥0};

(ii) x_{i,j} = x_{j,i} all i, j;
(iii) x_{i,j} ≤ x_{i,k} + x_{k,j} all i, j, k.

To each x⃗ = (x_{i,j})_{i,j∈ℕ} ∈ X(M) we may naturally associate a complete metric space S((x_{i,j})_{i,j∈ℕ}) with a dense subset (ax⃗ ; i)_{i∈ℕ} with d(a_{i}, a_{j}) = x_{i,j}. Conversely, if M is a complete separable metric space then by choosing (a_{i})_{i∈ℕ} to be any dense subset we may obtain some (x_{i,j})_{i,j∈ℕ} ∈ X(M) with S((x_{i,j})_{i,j∈ℕ}) isometric to M. Note that X(M) is a closed subset of ℝ^{ℕ×ℕ}, and thus is Polish in the subspace topology. We allow the possibility of x_{i,j} = 0, in which case we identify ax⃗ ; i and ax⃗ ; j; in this way we are able to represent finite metric spaces in X(M).

Now let X(K) be the set of (x_{i,j})_{i,j∈ℕ} ∈ X(M) ∈ [0, 1]^{ℕ×ℕ} which code compact metric spaces; note I specifically ask that each x_{i,j} be less than or equal to 1, but even with this restriction we may code all compact metric spaces up to homeomorphism. This is an F_{σδ} subspace of X(M), since compactness of S((x_{i,j})_{i,j∈ℕ}) amounts to the requirement that for all positive rationals q there is a finite set F ⊂ ℕ such that for all j ∈ ℕ there is some i ∈ F with x_{i,j} ≤ q.

For x⃗ = (x_{i,j})_{i,j∈ℕ}, y⃗ = (y_{i,j})_{i,j∈ℕ} ∈ X(K) write x⃗ ≅^{c} y⃗ if S(x⃗) and S(y⃗) are homeomorphic, x⃗ ≅^{i} y⃗ if isometric.

Lemma 4.20. ≅^{c} is Σ ∼_{1}^{1}.

Proof. Let (z_{i,j})_{i,j∈ℕ} be a sequence of non-negative reals. We need to first argue that the set of such z⃗ for which there is some fz⃗ : {ax⃗ ; i : i ∈ ℕ} → S(y⃗) such that

(i) d_{S(y⃗)}(ay⃗ ; i, fz⃗(ax⃗ ; j)) = z_{i,j} each i, j;

(ii) fz⃗ extends to a continuous function Fz⃗ : S(x⃗) → S(y⃗). is Borel. However the existence of an fz⃗ as indicated by (i) is transparently Borel, since

it amounts to verifying certain triangle inequalities along with inf_{i∈ℕ}(z_{i,j}) = 0 all j and that for all ϵ > 0 there is some i with z_{i,j} < ϵ, and hence that the supposed point fz⃗(ax⃗ ; j) is in the completion of metric space provided by y⃗. The further requirement that it extend to a continuous function on S(x⃗) amounts to only asserting that for all positive rational q there is some other positive rational r

such that (ax⃗ ; i, ax⃗ ; j) < r ⇒ (fx⃗(ax⃗ ; i), fx⃗(ax⃗ ; j)) < q.

Similar calculations show that the statement that z⃗ and and w⃗ code up inverse functions Fz⃗ : S(x⃗) → S(y⃗) and Fw⃗ : S(y⃗) → S(x⃗) is again Borel.

Exercise 4.21. (i) Use the proof of 4.17 to associate to each g⃗ ∈ G a compact metric space K(g⃗) so that

g⃗ − h⃗ ∈ Ĝ

if and only if

K(g⃗) ≅ K(h⃗).

(Hint: Build K(g⃗) like this: Begin with the compact metric space [0, 1]^{2} in ℝ^{2}. Add the the closed line segment from (0, 1/2) to (−1, 1/2) with the idea, as in the proof of 4.17, to distinguish the point (0, 1/2). At each I_{n,k} attach an isometric copy of [0, 2^{−n−|k|}]^{|g(n)−k|+2} × I_{n,k} with [0, 2^{−n−|k|}]^{|g(n)−k|+1} × {0} × I_{n,k} identified with our original rectangle I_{n,k}; so above each I_{n,k} rectangle there is a hypercube with dimension indicated by g⃗.)

(ii) Conclude that

G/Ĝ ≤_{B}≅^{c} .

Exercise 4.22. Let L be the language with a single binary relation <. Let Mod(LO) be the models in which < defines a linear ordering. Show that

≅|_{Mod(LO)} ≤_{B}≅^{c} .

(Given M in which < defines a linear ordering, choose open subsets of the open unit interval (0, 1) arranged in the order type indicated by M and so that their union U(M) is dense in [0, 1]. Then start defining K(M) by beginning with the compact metric space [0, 1]^{2} \ (U(M) × (0, 1)) but fastening around the boundary of the unit square a further compact metric space which will ensure that whenever π : K(M) ≅^{c} K(N) we must have π(0, 0) = (0, 0), π(1, 0) = (1, 0) and π[[0, 1] × {0}] = [0, 1] × {0}.)

Thus by the last two exercises, 4.16, and 4.10,

≅|_{Mod(L)} <_{B}≅^{c}

for any countable language L.

4.4. Remarks and connections

Perhaps we should not be surprised that there can be no result in the style of [24] for the classification of homeomorphisms of an arbitrary compact metric space. On the other hand, the homeomorphism group of the Cantor space, Hom(2^{ℕ}), can be realized as a closed subgroup of S_{∞}, and thus by 2.39 all its orbit equivalence relations admit classification by countable models. In particular then the the classification problem for homeomorphisms of 2^{ℕ} allows reduction to isomorphism on countable models.

Recently Riccardo Camerlo and Su Gao in [9] have showed the converse: For any countable language L we have that ≅ |_{Mod(L)} is ≤_{B} reducible to the orbit equivalence relation of Hom(2^{ℕ})’s self action by conjugation.

Although the equivalence relation of homeomorphism on compact metric spaces would seem to be hopelessly complicated, there is some very narrow regularity result. Kechris and Solecki have recently observed that the results of §2.5 of [6] imply ≅^{c} is Borel reducible to the orbit equivalence relation resulting from a continuous action of some Polish group on a Polish space – namely the shift action of the homeomorphism group of the Hilbert cube, Hom([0, 1]^{ℕ}), on F([0, 1]^{ℕ}).

While the classification problem for compact metric spaces up to homeomorphism verges on the intolerable, the classification up to isometry is nearly trivial. It follows from the Gromov-Hausdorff metric in [27] that ≅^{i} is smooth.

The method of proof is to place a pseudo metric on the collection K of all metric spaces whose underlying set is included in some specific set such as the reals. Firstly one defines the Hausdorff topology on the compact subsets of some specific compact metric space (K, d) – for C_{1}, C_{2} ⊂ K one can let

d_{H}^{K}(C_{1}, C_{2}) = sup {d(x_{1}, C_{2}) : x_{1} ∈ C_{1}} + sup {d(C_{1}, x_{2}) : x_{2} ∈ C_{2}}

where d(x, C) is (as usual) the infimum of {d(x, y) : y ∈ C}. For K_{1}, K_{2} ∈ K one then defines the Gromov-Hausdorff “distance” between them, d_{GH}(K_{1}, K_{2}), to be the infimum of

d_{H}^{K}(C_{1}, C_{2})

where K ranges over all possible compact metric spaces, and C_{1} and C_{2} range over compact subsets of K that are isometric to K_{1} and K_{2} respectively. It can be shown here that

d_{GH}(K_{1}, K_{2}) = 0

if and only if K_{1} and K_{2} are isometric, and the space K/ ∼ obtained from K – our space of compact spaces – by quotienting out by the equivalence relation

K_{1} ∼ K_{2} ⇔ d_{GH}(K_{1}, K_{2}) = 0

is in fact a compact metric space. All the key calculations can be conducted in the Borel context, and so by 2.27 one has that ≅^{i} is smooth.

CHAPTER 5 Infinite dimensional group representations

This chapter shows that the infinite dimensional representations of the free group on ℵ_{0} many generators are generically turbulent under the natural action of U_{∞}. The irreducible representations of this group are seen to refuse classification by countable structures.

§2.1 defined the infinite unitary group, U_{∞}, to be the group of isomorphisms of Hilbert space and noted it to be Polish in the natural topology.

Definition 5.1. For G be a countable group, let Rep(G, U_{∞}) be the space of all homomorphisms from G to U_{∞}. We refer to elements of this space as infinite dimensional representations. We view this as a subspace of the G-fold product of U_{∞},

∏_{G} U_{∞}.

We let U_{∞} act in the following way: For ρ ∈ Rep(G, U_{∞}) and T ∈ U_{∞} we define T · ρ by the requirement that at each g ∈ G

(T · ρ)(g) = T ∘ ρ(g) ∘ T^{−1}.

I will denote the corresponding orbit equivalence relation by ∼_{r} – so that ρ_{1} ∼_{r} ρ_{2} if there is some T with T · ρ_{1} = ρ_{2}. When context makes clear, I will just write ρ_{g} for ρ(g).

The notion of equivalence, ∼_{r}, expressed in this definition is the natural one: We want to say that two representations are essentially the same if they become literally the same after some unitary transformation of the Hilbert space.

In the next lemma it is convenient to note that sets of the form

{ρ : ||ρ_{g}(e) − f|| < δ}

form a subbasis for Rep(G, U_{∞}) as e, f range over the unit vectors of the Hilbert space, g ranges over G, δ ranges over positive reals. By elementary manipulations one can then obtain a subbasis whose members are of the form

{ρ : |⟨ρ_{g}(e), f⟩ − 1| < δ},

e, f unit vectors, g ∈ G, δ ∈ ℝ^{+}.

Lemma 5.2. Rep(G, U_{∞}) is a Polish U_{∞}-space.

Proof. It is a closed subset of the product space, ∏_{G} U_{∞}, and so since the class of Polish spaces is preserved under countable products and the formation of closed subspaces, Rep(G, U_{∞}) is Polish. Moreover the action of U_{∞} on this space is continuous, since given a subbasic open set of the form

U = {ρ : |⟨ρ_{g}(e), f⟩ − 1| < δ},

for elements e, f of the Hilbert space, and given T_{0} ∈ U_{∞}, and given τ such that T_{0} · τ ∈ U it is clear that for any T and ρ with T^{−1}(e), T^{−1}(f), and ρ_{g} ∘ T_{0}^{−1}(e) sufficiently close to T_{0}^{−1}(e), T_{0}^{−1}(f), and τ_{g} ∘ T_{0}^{−1}(e) respectively we have

⟨((T · ρ)(g))(e), f⟩ = ⟨ρ_{g}(T^{−1}(e)), T^{−1}(f)⟩

very close to

⟨τ_{g}(T_{0}^{−1}(e)), T_{0}^{−1}(f)⟩ = ⟨((T_{0} · τ)(g))(e), f⟩,

since ρ_{g} is an isometry.

Examples 5.3. Let {e_{i} : i ∈ ℕ} be an orthonormal basis for the Hilbert space.

(a) Let G = ℤ, s ∈ ℝ, and i ≠ j ∈ ℕ. Then we may define ρ ∈ Rep(ℤ, U_{∞}) by

ρ_{n}(e_{i}) = e_{i}cos(ns) + e_{j}sin(ns),
ρ_{n}(e_{j}) = −e_{i}sin(ns) + e_{j}cos(ns),

and

ρ_{n}(e_{k}) = e_{k}

for k ≠ i, j. Each ρ_{n} extends uniquely to the rest of the Hilbert space after being so defined on the basis.

(b) Now suppose G is a countable subgroup of S_{∞}. Then we could define ρ : G → U_{∞} by

ρ_{g}(e_{i}) = e_{g(j)},

and extend as above. Indeed, this is a proof that S_{∞} can be realized as a closed subgroup of the infinite unitary group.

(c) Suppose G is any countable group and (τ^{i})_{i∈ℕ} is some infinite sequence of representations. Then we may form a kind of direct product of these representations as follows: Given a partition of the basis {e_{k} : k ∈ ℕ} into infinite sets A_{i} = {e_{i,j} : i ∈ ℕ} we may define

ρ^{∞} : G → U_{∞}

by

ρ_{g}^{∞}(e_{i,j}) = ∑_{k∈ℕ} λ_{i,j,k}e_{i,k}

where

ρ_{g}^{i}(e_{j}) = ∑_{k∈ℕ} λ_{i,j,k}e_{k}.

In this way we obtain a new representation that mimics each of the old representations on a sequence of closed orthogonal subspaces of the Hilbert space.

Definition 5.4. ρ ∈ Rep(G, U_{∞}) is irreducible if there is no non-trivial Hilbert subspace which is invariant under {ρ_{g} : g ∈ G}.

So neither the examples in 5.3(a) or (c) are irreducible. In the first case the space {ce_{i} + de_{j} : c, d ∈ ℂ} is a closed invariant subspace, and in the second the closure of the span of any A_{j} provides an invariant subspace.

Notation 5.5. Let 𝔽_{∞} denote the free group on infinitely many generators. In future let us denote the generators by {v_{i} : i ∈ ℕ}.

For this group we obtain that there is a comeager set of irreducibles.

Lemma 5.6. ∀^{∗}ρ ∈ Rep(𝔽_{∞}, U_{∞})(ρ is irreducible).

Proof. Note first of all that for any two unit elements, e and f, of the Hilbert space, with ||e|| = ||f|| = 1 we may find a unitary operator T with T (e) = f. To see this, we may suppose that f = αe + α′e′ some e′ orthogonal to e where |α|^{2} +|α′|^{2} = 1. Now we can extend {e, e′} to an orthonormal basis {e, e′, f_{0}, f_{1}, ....} and then following 5.3(a) let

T (e) = αe + α′e′,
T (e′) = −α′e + αe′,

and

T (f_{k}) = f_{k}.

The point is that for any ρ ∈ Rep(𝔽_{∞}, U_{∞}) and k ∈ ℕ we have some ρ′ ∈ Rep(𝔽_{∞}, U_{∞}) with ρ′_{v_{n}} = ρ_{v_{n}} all n ≤ k but ρ′_{v_{k+1}} = T.

Thus if {ê_{i} : i ∈ ℕ} is a dense subset of the unit elements, for each i, j and q ∈ ℚ^{+} we have that the set of ρ ∈ Rep(𝔽_{∞}, U_{∞}) such that there exists n with

|⟨ρ_{v_{n}}(ê_{i}), ê_{j}⟩ − 1| < q

is open and dense in the space of representations. Any representation in the intersection of all these dense open sets will be irreducible.

Lemma 5.7. Every orbit in Rep(𝔽_{∞}, U_{∞}) is meager.

Proof. Fix τ, and we attempt to show [τ]_{U_{∞}} meager. It suffices to show that we may cover U_{∞} by open sets {V_{i} : i ∈ ℕ} such that for all i and for all open U_{0} ⊂ Rep(𝔽_{∞}, U_{∞}) there is some non-empty open U_{1} ⊂ U_{0} such that for all ρ ∈ U_{1} and for all T ∈ V_{i}

T · ρ ≠ τ.

Let {f_{i} : i ∈ ℕ} be a relatively dense subset of the unit ball in our Hilbert space. Let V_{i} be the set

{T ∈ U_{∞} : |⟨T (e_{0}), f_{i}⟩ − 1| < √(99/100)}.

Let U_{0} be a non-empty open set. After perhaps shrinking to a basic subset we may assume it has the form

U_{0} = {ρ : |⟨ρ_{v_{i}}(e_{j}), f_{n(i,j)}⟩ − 1| < δ_{i,j}∀i, j ≤ k}.

Let i ∈ ℕ.

Case (A) |⟨τ_{v_{k+1}}(f_{i}), f_{i}⟩| > 1/2.

Then let U_{1} = {ρ ∈ U_{0} : |⟨ρ_{v_{k+1}}(e_{0}), e_{0}⟩| < 1/10}.

Claim(1) U_{1} is not empty.

Proof of claim. Let σ ∈ U_{0}; this much is okay since we assumed U_{0} is nonempty. We next define ρ_{v_{i}} for each generator v_{i} ∈ 𝔽_{∞}. ρ_{v_{i}} = σ_{v_{i}} for i ≠ k + 1, but ρ_{v_{k+1}}(e_{0}) = e_{1}, and ρ_{v_{k+1}}(e_{1}) = e_{0}, and ρ_{v_{k+1}}(e_{n}) = e_{n} for n ≠ 0, 1. Since {e_{n} : n ∈ ℕ} is a basis, ρ_{v_{k+1}} extends to a unitary operator. By the defining properties of the free group ρ extends to a homomorphism from 𝔽_{∞} to U_{∞}. Since ρ agrees with σ on the set {v_{0}, v_{1}, ..., v_{k}}, we indeed have ρ ∈ U_{0}. (Claim(1)□)

Claim(2) If T ∈ V_{i} and ρ ∈ U_{1} then T · ρ ≠ τ.

Proof of claim. Express T^{−1}(f_{i}) as T^{−1}(f_{i}) = c_{1}e_{0} + c_{2}f where e_{0} and f are orthogonal. Note then that |c_{1}|^{2} + |c_{2}|^{2} = 1, and from T ∈ V_{i} we have |c_{1}|^{2} > 99/100, and hence |c_{2}| < 1/10. We then calculate that

|⟨(T · ρ)_{v_{k+1}}(f_{i}), f_{i}⟩| = |⟨T ∘ ρ_{v_{k+1}} ∘ T^{−1}(f_{i}), f_{i}⟩|

= |⟨ρ_{v_{k+1}}(T^{−1}(f_{i})), T^{−1}(f_{i})⟩| = |⟨ρ_{v_{k+1}}(c_{1}e_{0} + c_{2}f), c_{1}e_{0} + c_{2}f⟩|

≤ |⟨ρ_{v_{k+1}}(c_{1}e_{0}), c_{1}e_{0}⟩|+|⟨ρ_{v_{k+1}}(c_{1}e_{0}), c_{2}f⟩|+|⟨ρ_{v_{k+1}}(c_{2}f), c_{1}e_{0}⟩|+|⟨ρ_{v_{k+1}}(c_{2}f), c_{2}f⟩|

< 1/10 + 1/10 + 1/10 + 1/100 < 1/2,

and thus T · ρ ≠ τ, as required. (Claim(2)□) Case (B) |⟨τ_{v_{k+1}}(f_{i}), f_{i}⟩| ≤ 1/2. Then let U_{1} = {ρ ∈ U_{0} : |⟨ρ_{v_{k+1}}(e_{0}), e_{0}⟩| > 9/10}. Claim(3) U_{1} is not empty. Proof of claim. This is parallel to the proof of claim (1). We first choose σ ∈ U_{0}

and then define ρ ∈ Rep(𝔽_{∞}, U_{∞}) so that ρ_{v_{i}} = σ_{v_{i}} for i ≠ k + 1, but ρ_{v_{k+1}} is just left to be the identity – ρ_{v_{k+1}}(e_{n}) = e_{n} all n. (Claim(3)□)

Claim(4) If T ∈ V_{i} and ρ ∈ U_{1} then T · ρ ≠ τ.

Proof of claim. Again write T^{−1}(f_{i}) = c_{1}e_{0}+c_{2}f where e_{0} and f are orthogonal. We then calculate that

|⟨(T · ρ)_{v_{k+1}}(f_{i}), f_{i}⟩| = |⟨T ∘ ρ_{v_{k+1}} ∘ T^{−1}(f_{i}), f_{i}⟩|
= |⟨ρ_{v_{k+1}}(T^{−1}(f_{i})), T^{−1}(f_{i})⟩| = |⟨ρ_{v_{k+1}}(c_{1}e_{0} + c_{2}f), c_{1}e_{0} + c_{2}f⟩|

≥ |⟨ρ_{v_{k+1}}(c_{1}e_{0}), c_{1}e_{0}⟩|−|⟨ρ_{v_{k+1}}(c_{1}e_{0}), c_{2}f⟩|−|⟨ρ_{v_{k+1}}(c_{2}f), c_{1}e_{0}⟩|−|⟨ρ_{v_{k+1}}(c_{2}f), c_{2}f⟩|

> 99/100 − 1/10 − 1/10 − 1/100 > 9/10 − 3/10 > 1/2,

as required. (Claim(4)□)

To summarize, we have shown that for each i there is an open dense set O_{i} ⊂ Rep(𝔽_{∞}, U_{∞}) such that for all ρ ∈ O_{i} and T ∈ V_{i}

T · ρ ≠ τ.

For C the comeager set

⋂_{i∈ℕ} O_{i}

we have that for all T ∈ U_{∞} = ⋃_{i∈ℕ} V_{i}

T · ρ ≠ τ.

We are left with verifying part (iii) in the definition of generic turbulence.

Lemma 5.8. There is some ρ ∈ Rep(𝔽_{∞}, U_{∞}) whose orbit is dense and whose local orbits are each somewhere dense – that is to say, for each τ ∈ [ρ]_{U_{∞}} and open U containing τ and each open neighborhood V of the identity in the unitary group, O(τ, U, V) contains an open set in its closure.

Proof. Let {ρ^{i} : i ∈ ℕ} be a countable dense subset of Rep(𝔽_{∞}, U_{∞}). Let {A_{i} : i ∈ ℕ} be a partition of the basis of the Hilbert space into infinitely many infinite disjoint sets, and let H_{i} be the closed subspace generated by A_{i}, and let ρ|_{H_{i}} be naturally similar to ρ^{i} – that is to say, if A_{i} = {e_{i,j} : j ∈ ℕ}, then

ρ_{v_{k}}(e_{i,j}) = ∑_{n∈ℕ} λ_{n}e_{i,n}

where

ρ_{v_{k}}^{i}(e_{j}) = ∑_{n∈ℕ} λ_{n}e_{n}.

It should be clear that the orbit of ρ is dense, since given any open U ⊂ Rep(𝔽_{∞}, U_{∞}) and τ^{i} ∈ U we obtain T with T · τ ∈ U by interchanging e_{j} and e_{i,j} for l ≤ N for some sufficiently large N. The point is that ρ^{i} falls inside the basic open U in virtue of its behavior on {e_{j} : j ≤ N} for some finite N.

Now suppose that τ ∈ [ρ]_{U_{∞}}. Let U be an open neighborhood of τ and let V be an open neighborhood of the identity operator; by possibly restricting to some subsets we may assume that U has the form

U = {σ ∈ Rep(𝔽_{∞}, U_{∞}) : ∀i, j, k ≤ K(|⟨σ_{v_{i}}(e_{j}), e_{k}⟩ − ⟨τ_{v_{i}}(e_{j}), e_{k}⟩| < δ)},

and that V has the form

V = {T ∈ U_{∞} : ∀j ≤ K(|⟨T (e_{j}), e_{j}⟩ − 1| < δ)}

for the same finite K and small δ > 0. Let U_{1} be an open non-empty subset of U. It suffices to show that O(τ, U, V)∩U_{1} is non-empty. By restricting we may assume that there is some finite L ≥ K such that for any two σ, σ′ we have that if

⟨σ_{v_{i}}(e_{j}), e_{k}⟩ = ⟨σ′_{v_{i}}(e_{j}), e_{k}⟩

all i, j, k ≤ L then

σ ∈ U_{1} ⇔ σ′ ∈ U_{1}.

In other words, membership in U_{1} is a function of a representation’s behavior with respect to v_{i}, e_{j} just for i, j ≤ L.

We may assume that there is some large, but finite N, with

⟨τ_{v_{i}}(e_{j}), e_{n,l}⟩ = 0

all i, j ≤ L, n ∈ ℕ, and l > N. In fact we may assume τ = S · ρ where S|_{H_{l}} = id|_{H_{l}} all sufficiently large l, since

{T ∈ U_{∞} : ∃M∀l > M(T |_{H_{l}} = id|_{H_{l}})}

is a dense subgroup of U_{∞}.

Now by the assumption on the {ρ^{i} : i ∈ ℕ} we may choose some large m with A_{m} ⊂ {e_{n} : n > N} and ρ^{m} ∈ U_{1}. Note then that if we define T_{π/2} ∈ U_{∞} by the requirement that T_{π/2}(e_{m,i}) = e_{i} and T_{π/2}(e_{i}) = −e_{m,i} at i ≤ L but T_{π/2}(e_{j}) = e_{j} for e_{j} not among {e_{0}, e_{1}, ..., e_{L}} ∪ {e_{m,0}, e_{m,1}, ..., e_{m,L}} then at all i, j, k ≤ L we have

⟨(T_{π/2} · τ)_{v_{i}}(e_{j}), e_{k}⟩ = ⟨(T_{π/2} · (S · ρ))_{v_{i}}(e_{j}), e_{k}⟩
= ⟨ρ_{v_{i}}((T_{π/2} ∘ S)^{−1}(e_{j})), (T_{π/2} ∘ S)^{−1}(e_{k})⟩
= ⟨ρ_{v_{i}}(S^{−1}(T_{π/2}^{−1}(e_{j}))), S^{−1}(T_{π/2}^{−1}(e_{k}))⟩
= ⟨ρ_{v_{i}}(S^{−1}(e_{m,j})), S^{−1}(e_{m,k})⟩ = ⟨ρ_{v_{i}}(e_{m,j}), e_{m,k}⟩,

by assumption on S, which in turn equals

⟨ρ_{v_{i}}^{m}(e_{j}), e_{k}⟩

and therefore we find that T_{π/2} · τ ∈ U_{1}.

This much indicates our destination. If we can just show there is some way to get there staying inside U and V we will be done; we need show

T_{π/2} · τ ∈ O(τ, U, V).

Now for each φ ∈ [0, π/2] we define T_{φ} by

T_{φ}(e_{m,i}) = e_{i}sin(φ) + e_{m,i}cos(φ),
T_{φ}(e_{i}) = −e_{i,m}sin(φ) + e_{i}cos(φ),

for i ≤ L, and for e_{n} not in {e_{0}, e_{1}, ..., e_{L}} ∪ {e_{m,0}, e_{m,1}, ..., e_{m,L}} set

T_{φ}(e_{n}) = e_{n}.

In the case φ = π/2 this agrees with our definition of T_{π/2} above.

Claim. For all φ ∈ [0, 2π]

T_{φ} · ρ ∈ U.

Proof of claim. We calculate. Fix i, j, k ≤ K. Then

⟨(T_{φ} · τ_{v_{i}})(e_{j}), e_{k}⟩ = ⟨τ_{v_{i}}(T_{φ}^{−1}(e_{j})), T_{φ}^{−1}(e_{k})⟩
= ⟨τ_{v_{i}}(e_{j}cos(φ) + e_{m,j}sin(φ)), e_{k}cos(φ) + e_{m,k}sin(φ)⟩
= ⟨τ_{v_{i}}(e_{j}cos(φ)) + τ_{v_{i}}(e_{m,j}sin(φ)), e_{k}cos(φ) + e_{m,k}sin(φ)⟩;

by assumption that S|_{H_{m}} = id and S · ρ = τ we have

⟨τ_{v_{i}}(e_{j}), e_{m,k}⟩ = 0

and

⟨τ_{v_{i}}(e_{m,j}), e_{k}⟩ = ⟨e_{m,j}, τ_{(v_{i})^{−1}}(e_{k})⟩ = 0,

and thus

⟨τ_{v_{i}}(e_{j}cos(φ)) + τ_{v_{i}}(e_{m,j}sin(φ)), e_{k}cos(φ) + e_{m,k}sin(φ)⟩
= ⟨τ_{v_{i}}(e_{j}cos(φ)), e_{k}cos(φ)⟩ + ⟨τ_{v_{i}}(e_{m,j}sin(φ)), e_{m,k}sin(φ)⟩
= cos^{2}(φ)⟨τ_{v_{i}}(e_{j}), e_{k}cos⟩ + ⟨sin^{2}(φ)⟨ρ_{v_{i}}^{m}(e_{j}), e_{k}⟩.

Thus

|⟨(T_{φ} · τ_{v_{i}})(e_{j}), e_{k}⟩ − ⟨τ_{v_{i}}(e_{j}), e_{k}⟩|
= |cos^{2}(φ)⟨τ_{v_{i}}(e_{j}), e_{k}⟩ + sin^{2}(φ)⟨ρ_{v_{i}}^{m}(e_{j}), e_{k}⟩ − ⟨τ_{v_{i}}(e_{j}), e_{k}⟩|
≤ sin^{2}(φ)|⟨ρ_{v_{i}}^{m}(e_{j}), e_{k}⟩ − ⟨τ_{v_{i}}(e_{j}), e_{k}⟩| < sin^{2}(φ)δ,

as required. Since this holds for all i, j, k ≤ K we have membership of T_{φ} · τ in U. (Claim□)

But now if we choose some sufficiently large M we will have that T_{π/2M} ∈ V. Then letting τ^{0} = τ, τ^{i+1} = T_{π/2M} · τ^{i}, we obtain witnesses that τ^{M} = T_{π/2} · τ ∈ O(τ, U, V).

Theorem 5.9. The irreducible representations of 𝔽_{∞} considered up to ∼_{r} do not admit classification by countable structures.

Proof. From 5.6 there is an invariant dense G_{δ}, call it C_{0}, on which all the representations are irreducible. We obtain from 5.7 and 5.8 applied to 3.21 that there is an invariant dense G_{δ} set C_{1} ⊂ Rep(𝔽_{∞}, U_{∞}) on which the action is turbulent.

Then taking X = C_{0} ∩ C_{1} we have a Polish U_{∞}-space refusing classification by countable structures whose orbits consist of irreducible representations considered up to ∼_{r}.

Of course 5.9 also gives a proof that ∼_{r} for the irreducibles is not smooth for G = 𝔽_{∞}, but actually much stronger results of this form are known. Thoma in [79] shows that the countable groups for which the space of irreducible representations considered up to ∼_{r} is smooth are exactly those with an abelian subgroup of finite index.

CHAPTER 6 A generalized Scott analysis

§6.1 reworks the Scott analysis and presents applications to the analysis of the Borel complexity of orbits. In particular we use the make over of Scott to develop some simple cases of the Hjorth-Kechris-Louveau theory of Borel equivalence relations induced by continuous S_{∞} actions and to obtain that every orbit in a Polish S_{∞}-space is Borel. This fact actually holds in complete generality for Polish group actions, and in the exercises the reader can find a unrelated proof of the general case as well as a demonstration that even though every individual orbit must be Borel the orbit equivalence relation itself need not be a Borel subset of the product space.

§6.2 provides the heart of this chapter and generalizes Scott’s method to arbitrary Polish groups; the new idea is to embark on a transfinite analysis of the local orbits, as defined in §3. We define the class of GE-groups, which in chapter 7 will be proved to include all nilpotent and invariantly metrizable Polish groups, as well as all countable products of locally compact Polish groups, and show that the generalized Scott analysis leads to a characterization of when the action of a GE group allows classification by countable models; among other things we see that such a classification is possible if and only if there is no continuous reduction of a turbulent orbit equivalence relation and moreover when such classification is possible we can in fact find a reduction that admits a Borel inverse up to orbit equivalence and complete HC invariants can be calculated in a canonical manner.

§6.3 is nothing other than a counterexample: For non-GE groups Borel reduction to isomorphism on countable models and complete classification by HC invariants may possible even when the canonical invariants of §6.2 fail to be complete.

§6.4 is a tentative step towards trying to find complete invariants that lie outside the class of hereditarily countable sets. In some ways it represents a side line to the main direction of the book, and it is hard to escape the feeling that a more penetrating approach should be possible. On the other hand it does seem to constitute the best tool we currently have for the “fine analysis” of the Borel structure of orbits; in particular one can show that if every orbit is no worse than Π ∼_{α}^{0} then the overall equivalence relation can be no worse than Π ∼_{α+ω}^{0}.

There is a sense in which §6.1 is superfluous. The analysis of §6.2 is more general and the exercises at the end of §6.1 never resurface later in the text. However this preliminary section is important in giving a perspective; it underscores how S_{∞} actions are far better understood than those of arbitrary Polish groups.

6.1. A preliminary discussion of a specific case

The original Scott analysis was for the logic action at 2.37 of S_{∞} on Mod(L). This allows a harmless variation for general actions of S_{∞}, which while less intuitive than the original suggests the modifications needed for arbitrary Polish groups. I will only phrase the results for S_{∞}, but they hold in the context of arbitrary closed subgroups of the symmetric group. In any case, glancing ahead to 7.18 one sees that the orbit equivalence relation induced by a closed subgroup of the symmetric group will be ≤_{B} reducible to an appropriately chosen E_{S_{∞}}^{X}.

Definition 6.1. Let X be a Polish S_{∞}-space. Let (U_{l})_{l∈ℕ} be a basis of X, and let

V_{n} = {g ∈ S_{∞} : ∀i < n(g(i) = i)},

so that (V_{n})_{n∈ℕ} is a neighborhood basis at the identity consisting of clopen subgroups. For x ∈ X we then define φ_{α}(x, V_{n}) by simultaneous induction on the ordinal α.

φ_{0}(x, V_{n}) = {l : ∃x̂ ∈ U_{l} ⋂ V_{n} · x};
φ_{α+1}(x, V_{n}) = {⟨φ_{α}(x̂, V_{m}), m⟩ : x̂ ∈ V_{n} · x, m ≥ n};
for λ a limit, φ_{λ}(x, V_{n}) = {⟨α, φ_{α}(x, V_{n})⟩ : α < λ}.

The definition is not quite precise, since φ_{0}(x, V_{n}) depends not just on x and V_{n}, but the entire apparatus of the group action and the choice of the sequence (U_{l})_{l∈ℕ}. However a cure such as φ_{0}(x, V_{n}, (U_{l})_{l∈ℕ}, a, X) would hardly be a happy one.

Noting (V_{n})^{k} = V_{n} for all k ∈ ℤ we have {l : ∃x̂ ∈ U_{l} ⋂ V_{n} · x} = φ_{0}(x, X, V_{n}), and hence φ_{0}(x, V_{n}) = φ_{0}(x, X, V_{n}), in the sense of §3.2. The existence of neighborhood basis at the identity consisting of open subgroups is a key property of closed subgroups of S_{∞} and this hints at the direction that the generalized analysis must follow in §6.2.

The next lemma expresses that φ_{α}(x, V_{n}) indicates at least as much as φ_{β}(x, V_{n}) for all β ≤ α

Lemma 6.2. For all x_{1}, x_{2} ∈ X, β ≤ α

φ_{α}(x_{1}, V_{n}) = φ_{α}(x_{2}, V_{n}) ⇒ φ_{β}(x_{1}, V_{n}) = φ_{β}(x_{2}, V_{n}).

Proof. By transfinite induction on α starting with the base case β = α. Lifting the induction through limit stages is trivial, and so this just leaves us with showing that given the lemma at α we may ratchet up to α + 1.

Suppose then that φ_{α+1}(x_{1}, V_{n}) = φ_{α+1}(x_{2}, V_{n}). In particular we may find x̂_{2} ∈ V_{n} · x_{2} with

φ_{α}(x_{1}, V_{n}) = φ_{α}(x̂_{2}, V_{n}).

Now since φ_{α}(x̂_{2}, V_{n}) = φ_{α}(x_{2}, V_{n}) we can apply the inductive assumption.

Lemma 6.3. For all x ∈ X there is some γ^{∗}(x) < ω_{1} such that for all x_{1}, x_{2} ∈ [x]_{S_{∞}}, n ∈ ℕ

∃α < ω_{1}(φ_{α}(x_{1}, V_{n}) ≠ φ_{α}(x_{2}, V_{n})) ⇔ φ_{γ^{∗}(x)}(x_{1}, V_{n}) ≠ φ_{γ^{∗}(x)}(x_{2}, V_{n}).
Proof. For x_{1}, x_{2} ∈ [x]_{S_{∞}} and n ∈ ℕ set
x_{1} ≈_{n}^{∗} x_{2}

if x_{1} ∈ V_{n} · x_{2}. Note that this is an equivalence relation, [x]_{S_{∞}} has only countably many ≈_{n}^{∗} equivalence classes, and for α < ω_{1}

∀x_{1}, x_{2} ∈ [x]_{S_{∞}}(x_{1} ≈_{n}^{∗} x_{2} ⇒ φ_{α}(x_{1}, V_{n}) = φ_{α}(x_{2}, V_{n})).

So by cardinality considerations there is some γ_{n}(x) < ω_{1} such that for all α < ω_{1}

φ_{α}(x_{1}, V_{n}) ≠ φ_{α}(x_{2}, V_{n}) ⇒ φ_{γ_{n}(x)}(x_{1}, V_{n}) ≠ φ_{γ_{n}(x)}(x_{2}, V_{n}).

Then γ^{∗}(x) = sup{γ_{n}(x) : n ∈ ℕ} is as desired.

In fact γ^{∗}(x) above has the property that for any ordinal α and x_{1}, x_{2} in the orbit of x

φ_{α}(x_{1}, V_{n}) ≠ φ_{α}(x_{2}, V_{n}) ⇒ φ_{γ^{∗}(x)}(x_{1}, V_{n}) ≠ φ_{γ^{∗}(x)}(x_{2}, V_{n}).

Definition 6.4. For x ∈ X let γ^{∗}(x) be the least countable ordinal satisfying the conclusion of 6.3. Let φ_{x}^{∗} = φ_{γ^{∗}(x)+2}(x, V_{0}).

γ^{∗}(x) and φ_{x}^{∗} are invariants of the orbit equivalence relation, in that if g·x_{1} = x_{2} for some g ∈ S_{∞} then

γ^{∗}(x_{1}) = γ^{∗}(x_{2}),
φ_{x_{1}}^{∗} = φ_{x_{2}}^{∗}.

The second of these equalities arises from the choice of V_{0} = S_{∞}.

We also have that γ^{∗}(x) is indicated by φ_{x}^{∗}, in the following sense:

Lemma 6.5. If φ_{γ^{∗}(x)+2}(y, V_{0}) = φ_{x}^{∗} then for all y_{1}, y_{2} ∈ [y]_{S_{∞}}(= V_{0} · y) and n ∈ ℕ β < ω_{1}

φ_{γ^{∗}(x)}(y_{1}, V_{n}) = φ_{γ^{∗}(x)}(y_{2}, V_{n}) ⇒ φ_{β}(y_{1}, V_{n}) = φ_{β}(y_{2}, V_{n}).

Proof. By 6.2 we have this for β ≤ γ^{∗}(x) and it is trivial to show that if λ is a limit and 6.5 holds for all β = α < λ then it holds for β = λ. So instead let us assume that the lemma is true for β = α and attempt to lever up to the case β = α + 1. For this purpose, fix y_{1}, y_{2} ∈ [y]_{S_{∞}} with

φ_{γ^{∗}(x)}(y_{1}, V_{n}) = φ_{γ^{∗}(x)}(y_{2}, V_{n}).
By the assumption on y we may find x_{1}, x_{2} ∈ [x]_{S_{∞}} with
φ_{γ^{∗}(x)+1}(y_{1}, V_{n}) = φ_{γ^{∗}(x)+1}(x_{1}, V_{n}),
φ_{γ^{∗}(x)+1}(y_{2}, V_{n}) = φ_{γ^{∗}(x)+1}(x_{2}, V_{n}).

Then since

φ_{γ^{∗}(x)}(x_{1}, V_{n}) = φ_{γ^{∗}(x)}(x_{2}, V_{n})

it follows by definition of γ^{∗}(x) that

φ_{γ^{∗}(x)+1}(x_{1}, V_{n}) = φ_{γ^{∗}(x)+1}(x_{2}, V_{n}),
∴ φ_{γ^{∗}(x)+1}(y_{1}, V_{n}) = φ_{γ^{∗}(x)+1}(y_{2}, V_{n}).

So at once by definition

{⟨φ_{γ^{∗}(x)}(ŷ, V_{m}), m⟩ : ŷ ∈ V_{n}·y_{1}, m ≥ n} = {⟨φ_{γ^{∗}(x)}(ŷ, V_{m}), m⟩ : ŷ ∈ V_{n}·y_{2}, m ≥ n},

on which the inductive assumption produces

{⟨φ_{α}(ŷ, V_{m}), m⟩ : ŷ ∈ V_{n} · y_{1}, m ≥ n} = {⟨φ_{α}(ŷ, V_{m}), m⟩ : ŷ ∈ V_{n} · y_{2}, m ≥ n},

∴ φ_{α+1}(y_{1}, V_{n}) = φ_{α+1}(y_{2}, V_{n}).

It is important to observe that each of these φ_{α}(x, V_{n}) objects is an hereditarily countable set (assuming here the Kuratowski device for coding the pairing function – ⟨a, b⟩ = {{a, b}, b}). Moreover the invariants naturally level off with an appropriate number of iterations of the operation “a countable subset of”.

φ_{0}(x, V_{n}) is a point in 2^{ℕ}; φ_{1}(x, V_{n}) may be coded by a countable subset of 2^{ℕ}, since if we have some one to one s(·, ·, ...) : ℕ^{<ℕ} ↪ ℕ then we can represent φ_{1}(x, V_{n}) by

{{s(l, n̂) : l ∈ φ_{0}(x̂, Vn̂), n̂ ≥ n} : x̂ ∈ V_{n} · x};

φ_{2}(x, V_{n}) can be coded by a countable set of countable subsets of 2^{ℕ}, and so on. One usually defines the hereditarily countable sets to be the sets a for which the transitive closure

a ∪ ⋃ a ∪ ⋃ ⋃ a ∪ ...

is countable. Under this definition set theorists understand each natural number to be hereditarily countable, with the identification 0 = ∅, 1 = {∅}, 2 = {0, 1}, and onwards. Alternatively we could just choose to define the hereditarily countable sets to be the smallest class of sets containing the natural numbers and closed under the taking of countable subsets. In any case, the φ_{α}(x, V_{n}) invariants are all elements of HC, the class of hereditarily countable sets.

The next theorem is morally due to Dana Scott, who proved this much for X = Mod(L), some countable language L.

Theorem 6.6. For x, y ∈ X

xE_{S_{∞}}y ⇔ φ_{x}^{∗} = φ_{y}^{∗}.

Proof. As above, the forward direction ⇒ is trivial; so instead suppose that φ_{x}^{∗} = φ_{y}^{∗}.

We will find sequences of points

x_{0} = x, x_{1}, x_{2}, ... ∈ [x]_{S_{∞}},
y_{0} = y, y_{1}, y_{2}, ... ∈ [y]_{S_{∞}},

group elements

g_{0} = 1_{S_{∞}}, g_{1}, g_{2}, ...
h_{0} = 1_{S_{∞}}, h_{1}, h_{2}, ...

and natural numbers

l(0), l(1), l(2), ..., n(0), n(1), n(2), ...

such that at each i (1) x_{i} = g_{i} · x, y_{i} = h_{i} · y;

(2) x_{i}, y_{i} ∈ U_{l(i)}, d(U_{l(i)}) < 2^{−i} for i > 0, U_{l(i+1)} ⊂ U_{l(i)};
(3) φ_{γ^{∗}(x)}(x_{i}, V_{n(i)}) = φ_{γ^{∗}(x)}(y_{i}, V_{n(i)});
(4a) g_{i+1}g_{i}^{−1} ∈ V_{n(i)};
(4b) h_{i+1}h_{i}^{−1} ∈ V_{n(i)};
(5a) if i odd, then for all k < i we have g_{i}(k), g_{i}^{−1}(k) < n(i);
(5b) if i even, then for all k < i we have h_{i}(k), h_{i}^{−1}(k) < n(i);
(6) n(i) < n(i + 1).

Supposing for the moment that this is all done then we finish abruptly. By (4)-(6) we have that

g_{i} → g_{∞}

y_{0} x_{0} = x_{1} x_{2} = x_{3}  y_{1} = y_{2}  U_{n(1)} U_{n(2)} U_{n(3)}

Figure 6.1. The x_{i}’s and y_{i}’s drawing closer together

and

h_{i} → h_{∞}

for some g_{∞}, h_{∞} ∈ S_{∞}. By (1) and (2) we have

x_{i} → g_{∞} · x,
y_{i} → h_{∞} · y,

and so

g_{∞} · x = h_{∞} · y,

thereby establishing xE_{S_{∞}}y.

Now working to show that all this can be produced suppose inductively that we have x_{i}, y_{i}, g_{i}, h_{i}, l(i), n(i) for i < j, and seek x_{j}, y_{j}, g_{j}, h_{j}, l(j), n(j). For simplicity assume that j is even; the odd case is exactly symmetrical. First simply choose n(j) to be the sup of {n(j − 1) + 1, h_{j−1}(k), h_{j−1}^{−1}(k) : k < j}, let h_{j} = h_{j−1}, and let y_{j} = y_{j−1}. We take l(j) such that y_{j}(=_{df} y_{j−1}) ∈ U_{l(j)} ⊂ U_{l(j−1)} and d(U_{l(j)}) < 2^{−j}

On the x side there is more work.

Claim: φ_{γ^{∗}(x)+1}(x_{j−1}, V_{n(j−1)}) = φ_{γ^{∗}(x)+1}(y_{j−1}, V_{n(j−1)}).

Proof of claim: Suppose otherwise. Then since

{φ_{γ^{∗}(x)+1}(x′, V_{n}) : x′ ∈ [x]_{S_{∞}}, n ∈ ℕ} = {φ_{γ^{∗}(x)+1}(y′, V_{n}) : y′ ∈ [y]_{S_{∞}}n ∈ ℕ} there will be some x′ ∈ [x]_{S_{∞}} with

φ_{γ^{∗}(x)+1}(x′, V_{n(j−1)}) = φ_{γ^{∗}(x)+1}(y_{j−1}, V_{n(j−1)})
≠ φ_{γ^{∗}(x)+1}(x_{j−1}, V_{n(j−1)})

Then it follows from

φ_{γ^{∗}(x)}(x_{j−1}, V_{n(j−1)}) = φ_{γ^{∗}(x)}(y_{j−1}, V_{n(j−1)})

that

φ_{γ^{∗}(x)}(x′, V_{n(j−1)}) = φ_{γ^{∗}(x)}(x_{j−1}, V_{n(j−1)}),

with a contradiction to the definition of γ^{∗}(x). (Claim□) Thus we may choose x_{j} ∈ V_{n(j−1)} · x_{j−1} ∩ U_{l(j)} such that

φ_{γ^{∗}(x)}(x_{j}, V_{n(j)}) = φ_{γ^{∗}(x)}(y_{j}, V_{n(j)})(=_{df} φ_{γ^{∗}(x)}(y_{j−1}, V_{n(j)}).

Then fix g ∈ V_{n(j−1)} with g · x_{j−1} = x_{j} and let g_{j} = gg_{j−1}.

And thus the construction continues it way through all j ∈ ℕ.

We then have that for continuous S_{∞} actions complete invariants can be assigned in HC, the collection of all hereditarily countable sets. The precise same argument works for any Polish group with a neighborhood basis at the identity consisting of clopen subgroups; it was recorded in [4] that these are up to isomorphism exactly the closed subgroups of S_{∞}.

The Scott invariants can be used in the sharp analysis of S_{∞} orbits. To mine into those details we first need some observations regarding the Vaught transforms from 2.51.

Lemma 6.7. Let H be a Polish group, Y a Polish H-space.

(a) If g ∈ H and y ∈ Y then

g · y ∈ B^{ΔV} ⇔ y ∈ B^{ΔV g}.

(b) If g ∈ H and y ∈ Y then

y ∈ (g^{−1} · B)^{ΔV} ⇔ y ∈ B^{gΔV} .

(c) If B ⊂ Y is invariant, in the sense that h · B = B for all h ∈ H then B^{ΔH} = B^{∗H} = B.

Proof. For (a), note that if there is a comeager subset C ⊂ W where W ⊂ V open with

∀h ∈ C(h · (g · y) ∈ B),

then

∀h ∈ Cg(h · y ∈ B),

and Cg is comeager in W g ⊂ V g since H acts by homeomorphism on itself. Thus we have g · y ∈ B^{ΔV} implies y ∈ B^{ΔV g}.

By the same argument (g^{−1} · (g · y)) ∈ B^{ΔV g} implies g · y ∈ B^{ΔV gg^{−1}} = B^{ΔV}. The proof of (b) resembles (a). Then (c) follows from (b).

Lemma 6.8. H, Y as in 6.7. For {W_{n} : n ∈ ℕ} a basis of H, W open in H, and {B_{n} : n ∈ ℕ} a sequence of Borel sets in Y,

(⋃_{ℕ} B_{n})^{ΔW} = ⋃{(B_{n})^{∗W_{m}} : m, n ∈ ℕ, W_{m} ⊂ W, W_{m} ≠ ∅},

(⋂_{ℕ} B_{n})^{∗W} = ⋂{(B_{n})^{ΔW_{m}} : m, n ∈ ℕ, W_{m} ⊂ W, W_{m} ≠ ∅}.

Proof. First for (⋃_{ℕ} B_{n})^{ΔW}. Since for B_{n} is Borel, y ∈ Y, and the set
{h ∈ W : h · y ∈ B_{n}}

is Borel. Hence

{h ∈ W : h · y ∈ ⋃_{n∈ℕ} B_{n}}

is non-meager if and only if there is some n with {h ∈ W : h·y ∈ B_{n}} non-meager, if and only if there is some W_{m} ⊂ W open and non-empty with {h ∈ W_{m} : h·y ∈ B_{n}} relatively comeager in W_{m}.

The lemma for (⋂_{ℕ} B_{n})^{∗W} follows by appealing to the first part of the proof to obtain

Y \ ((⋂_{ℕ} B_{n})^{∗W}) = (⋃_{ℕ} Y \ B_{n})^{ΔW} =

⋃{(Y \ B_{n})^{∗W_{m}} : m, n ∈ ℕ, W_{m} ⊂ W, W_{m} ≠ ∅}.

Lemma 6.9. H, Y, W as in 6.6. O open in Y implies O^{ΔW} open; C closed in Y implies C^{∗W} closed; D ⊂ Y a G_{δ} implies D^{∗W} ∈ G_{δ}; F ⊂ Y an F_{σ} implies that F^{ΔW} ∈ F_{σ}; D in G_{δσ} implies D^{ΔW} ∈ G_{δσ} (and more generally B ⊂ Y in Σ ∼_{α}^{0} implies B^{ΔW} ∈ Σ ∼_{α}^{0} while B ∈ Π ∼_{α}^{0} implies B^{∗W} ∈ Π ∼_{α}^{0}).

Proof. Continuity of the action at once implies that O^{ΔW} is open for any open W ⊂ H. Then the rest of the lemma follows by applying 6.8 as many times as necessary.

Lemma 6.10. Let X, (U_{l})_{l∈ℕ}, (V_{l})_{l∈ℕ} be as in 6.1, x, y ∈ X.

(i) If O ⊂ X open, n ∈ ℕ, φ_{0}(x, V_{n}) = φ_{0}(y, V_{n}), then

x ∈ O^{ΔV_{n}} ⇔ y ∈ O^{ΔV_{n}}.

(ii) If D ⊂ X an F_{σ}, φ_{1}(x, V_{n}) = φ_{1}(y, V_{n}), then

x ∈ D^{ΔV_{n}} ⇔ y ∈ D^{ΔV_{n}}.

(iii) More generally B ∈ Σ ∼_{m+1}^{0}, φ_{m}(x, V_{n}) = φ_{m}(y, V_{n}), implies

x ∈ B^{ΔV_{n}} ⇔ y ∈ B^{ΔV_{n}}.

Proof. (i) is a consequence of the definitions and the continuity of the action. For (ii) write B = ⋃_{ℕ} F_{i}, each F_{i} closed. Note then that there is some open V ⊂ V_{n}, i ∈ ℕ, V ≠ ∅ with x ∈ F_{i}^{ΔV} if and only if there is some x̂ ∈ V_{n} · x and Vn̂ ⊂ V_{n} with x̂ ∈ F_{i}^{∗Vn̂}, and so (ii) follows from (i). The general case at (iii) marches up the natural numbers by an appeal to mathematical induction.

With some minor notational regearing at the first limit this continues for all countable ordinals: If

B ∈ Σ ∼_{α}^{0},

α ≥ ω and φ_{α}(x, V_{n}) = φ_{α}(y, V_{n}), then

x ∈ B^{ΔV_{n}} ⇔ y ∈ B^{ΔV_{n}}.

Note that if E_{S_{∞}}^{X} is say F_{σδ} in X × X, and so that in particular for every x ∈ X

[x]_{S_{∞}} = ([x]_{S_{∞}})^{∗S_{∞}} ∈ F_{σδ},

then x ↦ φ_{2}(x, S_{∞}) is a complete invariant for the orbit equivalence relation. So from the coding mentioned after 6.5, E_{S_{∞}}^{X} is immediately seen to be reducible to F_{3} on 2^{ℕ×ℕ×ℕ}, isomorphism on codes for countable sets of countable subsets of 2^{ℕ}. Actually by [41]:

Theorem 6.11. (Hjorth, Kechris, Louveau) Let X be a Polish S_{∞} space with the orbit equivalence relation E_{S_{∞}} an F_{σδ} subset of X × X. Then E_{S_{∞}} ≤_{B} F_{2}.

Proof. Fix (U_{l})_{l∈ℕ}, (V_{l})_{l∈ℕ} as before. Fix x ∈ X. Let τ_{x} be the topology generated by taking the union of {{z : l ∉ φ_{0}(z, V_{n})} : l, n ∈ ℕ}, {U_{l} : l ∈ ℕ}, and {{z : φ_{0}(z, V_{n}) = φ_{0}(x̂, V_{n})} : n ∈ ℕ, x̂ ∈ [x]_{S_{∞}}} as a subbasis. Consider the space (X, τ_{x}) – by which I mean the space X equipped with τ_{x} as a topology.

Claim(I): (X, τ_{x}) is a Polish space.

Proof of claim. First let (C_{i})_{i∈ℕ} enumerate the closed sets of the form

{z : l ∉ φ_{0}(z, V_{n})}

for various l, n ∈ ℕ. We then form a sequence of topologies starting with τ_{0} = τ and then taking

τ_{i+1} = τ_{i} ⋃{U ∩ C_{i} : U ∈ τ_{i}}.

Each τ_{i} is Polish by 2.3(ii). Then let τ_{∞} be the topology with basis ⋃_{i∈ℕ} τ_{i}. By 2.3(iii) this is Polish.

We can go for another round, observing that each set of the form

{z : φ_{0}(z, V_{n}) = φ_{0}(x̂, V_{n})}

is now τ_{∞}-closed. Let (C_{∞+i})_{i∈ℕ} enumerate these sets. Once more set

τ_{∞+i+1} = τ_{∞+i} ∪ (⋃{U ∩ C_{∞+i} : U ∈ τ_{∞+i}})

and once more use 2.3(ii) to see that this is Polish. Again we can use 2.3(iii) to obtain that τ_{∞+∞} – the topology we want – is Polish. (Claim(I)□)

Let C_{x} be the closure of [x]_{S_{∞}} in τ_{x}. The pair (τ_{x}, C_{x}) is an invariant of the orbit [x]_{S_{∞}}. The next claim shows it to be a complete invariant.

Claim(II): ∀^{∗}y ∈ (C_{x}, τ_{x})(yE_{S_{∞}}x).

Proof of Claim: Let

[x]_{S_{∞}} = ⋂_{i∈ℕ} ⋃_{j∈ℕ} B_{i,j}

with each B_{i,j} closed. Note then that

[x]_{S_{∞}} = (⋂_{i∈ℕ} ⋃_{j∈ℕ} B_{i,j})^{∗S_{∞}}

= ⋂{⋃{B_{i,j}^{∗W} : j ∈ ℕ, W ⊂ V, basic open ≠ ∅} : i ∈ ℕ, V ⊂ S_{∞}, basic open ≠ ∅}.

Then for each i and V ⊂ S_{∞} basic open non-empty we have that the set O_{V,i} of y ∈ C_{x} such that there exists j ∈ ℕ and n ∈ ℕ and g ∈ S_{∞} and x̂ ∈ [x]_{S_{∞}} such that

g · x̂ ∈ B_{i,j}^{∗V_{n}},
V_{n}g ⊂ V,

and

φ_{0}(y, V_{n}) = φ_{0}(x̂, V_{n})

is dense open in (C_{x}, τ_{x}). Moreover for any y ∈ O_{V,i} and g, j, n as above

g · y ∈ B_{i,j}^{∗V_{n}}

by 6.10 and hence

y ∈ B_{i,j}^{∗V_{n}g},

with V_{n}g ⊂ V. But then for y meeting all these open dense O_{V,i}

y ∈ ⋂{⋃{B_{i,j}^{∗W} : j ∈ ℕ, W ⊂ V, basic open ≠ ∅} : i ∈ ℕ, V ⊂ S_{∞}, basic open ≠ ∅}

as required. (Claim(II)□) Claim(III): For x, y ∈ X

φ_{1}(x, V_{0}) = φ_{1}(y, V_{0}) ⇒ (C_{x}, τ_{x}) = (C_{y}, τ_{y}).

Proof of claim: Clearly φ_{1}(x, V_{0}) = φ_{1}(y, V_{0}) implies τ_{x} = τ_{y}. Note that for any x_{0} ∈ [x]_{S_{∞}} a neighborhood basis of the point is given by {φ_{0}(x_{0}, V_{n}) : n ∈ ℕ}. The assumption φ_{1}(x, V_{0}) = φ_{1}(y, V_{0}) then gives that the orbits meet the same τ_{x}-open sets. (Claim(III)□)

Since the φ_{1}(x, V_{0}) may be thought of as a countable subset of 2^{ℕ} we are done.

In fact this is precise, in the sense that E_{S_{∞}}^{X} ≤_{B} F_{2} if and only if there is some F_{σδ} equivalence relation F with E_{S_{∞}}^{X} ≤_{B} F. Thus we can characterize the equivalence relations induced by S_{∞} that are potentially F_{σδ}. More generally:

Theorem 6.12. (Hjorth, Kechris, Louveau) Let X be a Polish-S_{∞} space. Then there is a Π ∼_{n+1}^{0} equivalence relation F with E_{S_{∞}}^{X} ≤_{B} F if and only if E_{S_{∞}}^{X} ≤_{B} F_{n}.

Most of the further ideas needed for the proof are captured in:

Lemma 6.13. There is a Π ∼_{4}^{0} equivalence relation F with F_{3} ≤_{B} F.

Proof. To each x ∈ 2^{ℕ×ℕ×ℕ} associate y_{x} ∈ 2^{ℕ×ℕ×ℕ} with the provision that

{y_{x}(n, m, ·) : n, m ∈ ℕ} = {x(n, m, ·) : n, m ∈ ℕ}

and that for all x(n, m, ·) and n̂ ∈ ℕ

x(n, m, ·) ∈ {y_{x}(n̂, 2m̂, ·) : m̂ ∈ ℕ}

if and only if

x(n, m, ·) ∈ 2^{ℕ} \ {y_{x}(n̂, 2m̂ + 1, ·) : m̂ ∈ ℕ}

if and only if

x(n, m, ·) ∈ {x(n̂, m̂, ·) : m̂ ∈ ℕ}.

(Here make the minor supposition that for all n̂ there exists some n and m with x(n, m, ·) not an element of {x(n̂, m̂, ·) : m̂ ∈ ℕ}.) So on even coordinates y_{x}(n̂, 2m̂, ·) encodes the elements of {x(n̂, m, ·) : m ∈ ℕ} whilst on the odd y_{x}(n̂, 2m̂ + 1, ·) indicates the non-members of {x(n̂, m, ·) : m ∈ ℕ}.

This granted,

{y_{x}(n, m, ·) : m ∈ ℕ} = {yx̂(n̂, m, ·) : m ∈ ℕ}

is Π ∼_{2}^{0} in the presence of the assumption

{x(n, m, ·) : n, m ∈ ℕ} = {x̂(n, m, ·) : n, m ∈ ℕ},

since

{y_{x}(n, m, ·) : m ∈ ℕ} = {yx̂(n̂, m, ·) : m ∈ ℕ}

if and only if for all m, m̂ ∈ ℕ

y_{x}(n, 2m, ·) ≠ yx̂(n̂, 2m̂ + 1, ·).

Exercise 6.14. Let X be a Polish S_{∞}-space. Show by transfinite induction on α < ω_{1} that for all x_{0} ∈ X, and n ∈ ℕ the set

{x ∈ X : φ_{α}(x, V_{n}) = φ_{α}(x_{0}, V_{n})}

is Borel. Use 6.6 to conclude that [x_{0}]_{S_{∞}} is Borel.

The same conclusion is true in complete generality.

Exercise 6.15. (Compare [37]; G_{x,x}^{α} below coincides with G_{x}^{α} there.) Let G be an arbitrary Polish group and X a Polish G-space. Fix d a complete metric on G and B a countable basis of G. For x, y ∈ X let

G_{x,y} = {g ∈ G : g · x = y};

so this is nonempty if and only if xE_{G}y. Let G_{x,y}^{0} be the set of V ∈ B for which there exists g_{i} ∈ V

g_{i} · x → y;

then continue the definitions through the countable ordinals, with G_{x,y}^{α+1} being the set of V ∈ G_{x,y}^{α} such that for all n ∈ ℕ there exists some W ∈ G_{x,y}^{α} with

\overline{W} ⊂ V
d(W) < 2^{−n};

at λ a limit take

G_{x,y}^{λ} = ⋂_{α<λ} G_{x,y}^{α}.

(i) For any x, y there is some δ(x, y) < ω_{1} for which G_{x,y}^{δ(x,y)} = G_{x,y}^{δ(x,y)+1}.

(ii) For all V ∈ B we have G_{x,y} ∩ V ≠ ∅ ⇔ V ∈ G_{x,y}^{δ(x,y)}.

(iii) For each α < ω_{1} and V ∈ B the set {(x, y) ∈ X × X : V ∈ G_{x,y}^{α}} is Borel. (iv) For each V ∈ B and x, y ∈ X there exists a countable ordinal δ(V, x) so that

for any α > δ(V, x) there is a comeager set of g ∈ G with (V ∈ G_{x,g·y}^{α} ⇒ V ∈ G_{x,g·y}^{α+1}). (v) Let δ(x) = sup{δ(V, x) : V ∈ B}. For all x, y ∈ X

xE_{G}y

if and only if

∀^{∗}g ∈ G(G_{x,g·y}^{δ(x)} ≠ ∅).

(vi) Thus [x]_{G} is Borel.

Exercise 6.16. Show by induction on α that there is a Polish S_{∞} space such that for all α < ω_{1} there are x, y ∈ X for which

φ_{α}(x, V_{0}) = φ_{α}(y, V_{0})

without xE_{S_{∞}}y. Part (i) addresses the successor case of this induction. (i) Let M_{0}, M_{1} ∈ Mod(L) and suppose

φ_{α}(M_{0}, V_{0}) = φ_{α}(M_{1}, V_{0})

but

M_{0} ≅ M̸_{1}.

Let ≤ be a new binary relation and L′ be the language generated by ≤ and L. For each x ∈ 2^{ℤ} let N_{x} ∈ Mod(L′) be a model that arranges disjoint copies of M_{0} and M_{1} in a ℤ chain as indicated by x:

(a) ≤^{N_{x}} is a prelinear ordering of N_{x} (so that a ≤ b ≤ c implies that a ≤ c and any two elements are ≤ comparable);

(b) the ≡-equivalence classes of N_{x} defined by

a ≡ b ⇔_{df} a ≤^{N_{x}} b ≤^{N_{x}} a

are arranged in order type ℤ;

(c) for i ∈ ℤ, the ith ≡-equivalence class is isomorphic to M_{0} if x(i) = 0 and M_{1} if x(i) = 1.

If x, y ∈ 2^{ℤ} have [x]_{ℤ} and [y]_{ℤ} dense in the shift action of ℤ on 2^{ℤ} then

φ_{α+1}(N_{x}, V_{0}) = φ_{α+1}(N_{y}, V_{0}).

(ii) For each α < ω_{1} we can find a countable language L and M, M′ ∈ Mod(L) with

φ_{α}(M, V_{0}) = φ_{α}(M′, V_{0})

but

M ≅ M̸′.

(iii) These examples may be conjoined into a single countable language L_{∞} so that for each α < ω_{1} there is M ∈ Mod(L_{∞}) with

γ(M) > α.

The next exercise can be viewed as a topological formulation of an argument in [69]; it is related to the omitting types theorem for L_{ω_{1}ω}, as presented in [59], but the reformulation assumes no familiarity with infinitary logic.

Exercise 6.17. Suppose that X is a Polish S_{∞} space and x ∈ X with

[x]_{S_{∞}} ∈ Π ∼_{α}^{0}.

Let (X, φ_{α+1}(x, V_{0})) be the set of all y ∈ X with

φ_{α+1}(x, V_{0}) = φ_{α+1}(y, V_{0})

equipped with the topology generated by basic open sets of the form

{z : φ_{α}(x′, V_{n}) = φ_{α}(z, V_{n})}

some x′ ∈ [x]_{S_{∞}} and n ∈ ℕ.

(i) The Borel complexity of [x]_{S_{∞}} implies that (X, φ_{α+1}(x, V_{0})) = [x]_{S_{∞}}.

(ii) In the above topology (X, φ_{α+1}(x, V_{0})) is a Polish S_{∞}-space.

(iii) For each n let E_{n} be the orbit equivalence relation on (X, φ_{α+1}(x, V_{0})) induced by the continuous action of the clopen subgroup V_{n} of S_{∞}. 2.65 implies for each basic open

U = {z : φ_{α}(x′, V_{n}) = φ_{α}(z, V_{n})}

there is some m > n and x′′ ∈ [x]_{S_{∞}} so that

∀^{∗}y_{0}, y_{1} ∈ {z : φ_{α}(x′′, V_{m}) = φ_{α}(z, V_{m})} ⊂ U

we have

y_{0}E_{n}y_{1}.
(iv) For a comeager set of y in (X, φ_{α+1}(x, V_{0}))
γ^{∗}(y) ≤ α + 1

and hence

γ^{∗}(x) ≤ α + 1.

(Hint: Applying Kuratowski-Ulam we may find a comeager set y ∈ (X, φ_{α+1}(x, V_{0})) such that for all n and ∀^{∗}g ∈ S_{∞} there is some x′′ and m ≥ n witnessing (iii) above for g · y with

φ_{α}(g · y, V_{m}) = φ_{α}(x′′, V_{m}).)

(v) Thus by 6.16 it is possible to have a Polish S_{∞}-space X where the orbits are unbounded in Borel complexity, and in particular E_{S_{∞}}^{X} is non-Borel.

Exercise 6.18. Let X be a Polish S_{∞}-space. Let L be the language generated by a single binary relation R.

(i) For α < ω_{1} there is a Borel map θ_{α} : X → Mod(L) such that for all x ∈ X the structure θ_{α}(x) is isomorphic to

(TC({φ_{β}(x_{0}, V_{n}) : x_{0} ∈ [x]_{S_{∞}}, n ∈ ℕ, β < α}), ∈).

(ii) For each α < ω_{1}, the set of x ∈ X with γ^{∗}(x) = α is Borel.

(iii) Mimicking the steps (i)-(v) of 6.15 we obtain that for each Borel f : Y → X there is some δ < ω_{1} such that

∀^{∗}y ∈ Y (γ^{∗}(f(y)) < δ.

(iv) Thus choosing θ_{α} as in (i) for each α < ω_{1}, the resulting map

X → Mod(L)
x ↦ θ_{γ^{∗}(x)+2}(x)

is a universally Baire measurable reduction of E_{S_{∞}}^{X} to ≅|_{Mod(L)}.

Exercise 6.19. (See §2.7 [4]) Let X be a Polish S_{∞}-space with basis (U_{l})_{l∈ℕ} and let L be the language with infinitely many relations (R_{l,k})_{l∈ℕ} of arity k for each k ≥ 1. That is to say, each R_{n,1} will be a unary predicate for each n, R_{n,2} will be binary relation, R_{n,3} will be relation on triples and so on.

For n_{0}, n_{1}, ..., n_{k−1} a finite string of natural numbers, let

B_{n_{0},n_{1},...,n_{k−1}} = {g ∈ S_{∞} : g(n_{i}) = i all i < k}.

For x ∈ X define M(x) ∈ Mod(L) by the specification

M(x) ⊧ R_{l,k}(n_{0}, ..., n_{k−1})

if and only if

∀g ∈ B_{n_{0},n_{1},...,n_{k−1}}(g · x ∈ \overline{U_{l}}).

Then x ↦ M_{x} witnesses that E_{S_{∞}}^{X} ≤_{B}≅|_{Mod(L)}.

Exercise 6.20. Following the proof of 6.6 we have that if x_{0}, x_{1} ∈ [x]_{S_{∞}} with

φ_{γ^{∗}(x)}(x_{0}, V_{n}) = φ_{γ^{∗}(x)}(x_{1}, V_{n}),

then there is g ∈ V_{n} with

g · x_{0} = x_{1}.

The aim of the next section is to extend 6.7 to arbitrary Polish groups, though with a certain disappointment, since we now cannot demand that there are complete invariants in HC for arbitrary Polish groups. One compromise might be to consider invariants from a class more general than HC. Instead we will forgo the requirement that the invariants be complete, asking only that if any complete invariants in HC can be found then the some analogue of x ↦ φ_{x}^{∗} will provide the reduction. While this fails for complicated groups, it is true for a class of groups much broader than abelian, and the generalized Scott analysis of §6.2 will ultimately provide a canonical transfinite process that provably reduces an orbit equivalence relation to id(HC) if and only if any such reasonable reduction exists.

6.2. The general case

Definition 6.21. Let G be a Polish group and X be a Polish G-space. Fix (U_{n})_{n∈ℕ} a basis for X with U_{0} = X. Let (V_{n})_{n∈ℕ} be a neighborhood basis at 1_{G} with V_{0} = G and each V_{n} = V_{n}^{−1}. Let G_{0} = {g_{i} : i ∈ ℕ} be a countable dense subgroup of G and assume that the above bases are suitably invariant, in that for all i, j there are n, m ∈ ℕ such that

g_{i}V_{j}g_{i}^{−1} = V_{m},
g_{i} · U_{j} = U_{n}.

Recalling the definition of the local orbit O(x, U, V) from 3.15, for x ∈ X, U ⊂ X, V ⊂ G both open, α ≤ ω_{1}, we define φ_{α}(x, U, V) by induction on α:-

(1) φ_{0}(x, U, V) = {l : O(x, U, V) ∩ U_{l} ≠ ∅};

(2) φ_{α+1}(x, U, V) equals the set of all
⟨φ_{α}(x̂, U_{n}, V_{m}), n, m⟩
for x̂ ∈ O(x, U, V), U_{n} ⊂ U, V_{m} ⊂ V;
(3) for λ a limit, set φ_{λ}(x, U, V) = {⟨φ_{α}(x, U, V), α⟩ : α < λ}.

Lemma 6.22. (X, G, (U_{n})_{n∈ℕ}, (V_{n})_{n∈ℕ} as directly above.) For all x ∈ X, n, m ∈ ℕ there exists γ(x, n, m) < ω_{1} such that for all x_{0}, x_{1} ∈ [x]_{G}

∃α < ω_{1}(φ_{α}(x_{0}, U_{n}, V_{m}) ≠ φ_{α}(x_{1}, U_{n}, V_{m}))

implies

∃α ≤ γ(x, n, m)(φ_{α}(x_{0}, U_{n}, V_{m}) ≠ φ_{α}(x_{1}, U_{n}, V_{m})).

Proof. Fixing x, for each g, ĝ ∈ G with g · x, ĝ · x ∈ U_{n} set g ≈_{n,m}^{x} ĝ if there is some k ∈ ℕ and group elements g_{0}, g_{1}, ..., g_{k−1} ∈ V_{m} and h_{0}, h_{1}, ...h_{k} ∈ G such that:

h_{0} = g, h_{k} = ĝ, each h_{i+1} = g_{i}h_{i};
each g_{i} ∈ V_{m};
each h_{i} · x ∈ U_{n}.

This is an open equivalence relations on G and hence has at most countably many equivalence classes. It is also true that it provides congruence with respect to the functions g ↦ φ_{α}(g · x, U_{n}, V_{m}), in the sense that for any α < ω_{1} and any g ≈_{n,m}^{x} ĝ

φ_{α}(g · x, U_{n}, V_{m}) = φ_{α}(ĝ · x, U_{n}, V_{m}).

Now for any two equivalence classes [g]_{n,m}^{x}(=_{df} {h ∈ G : h ≈_{n,m}^{x} g}), [ĝ]_{n,m}^{x} we can let γ([g]_{n,m}^{x}, [ĝ]_{n,m}^{x}) be the least α such that φ_{α}(g · x, U_{n}, V_{m}) ≠ φ_{α}(ĝ · x, U_{n}, V_{m}) if it exists (0 if not). Then

γ(x, n, m) = sup{γ([g]_{n,m}^{x}, [ĝ]_{n,m}^{x}) : g, ĝ ∈ G, g · x, ĝ · x ∈ U_{n}}

is countable since we are taking the supremum over at most ℵ_{0} many equivalence classes.

Definition 6.23. Let γ(x, n, m) be least as in the conclusion of the lemma 6.22, and set γ(x) = sup{γ(x, n, m) : n, m ∈ ℕ}. Let φ_{x} = φ_{γ(x)+2}(x, X, G).

Lemma 6.24. x_{1}E_{G}^{X}x_{2} implies γ(x_{1}) = γ(x_{2}), and φ_{x_{1}} = φ_{x_{2}}.

Proof. For the first of these statements we note that the structure of the definitions gives at each n and m we have γ(x_{1}, n, m) = γ(x_{2}, n, m). The second then follows since O(x_{1}, X, G) = O(x_{2}, X, G).

Lemma 6.25. For all α < ω_{1} we have for all x, y ∈ X, n, m ∈ ℕ that
φ_{α+1}(x, U_{n}, V_{m}) = φ_{α+1}(y, U_{n}, V_{m})

implies

φ_{α}(x, U_{n}, V_{m}) = φ_{α}(y, U_{n}, V_{m}).

Proof. By transfinite induction: Suppose instead that this fails and let α be least for which there is some x, y, n, m providing a counterexample. First consider the case that α is a successor ordinal, so α = β + 1 for some β < ω_{1}. Then

{⟨φ_{α}(x̂, Un̂, Vm̂), n̂, m̂⟩ : x̂ ∈ O(x, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}
= {⟨φ_{α}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}

by definition of φ_{α+1}. Hence

{⟨φ_{β}(x̂, Un̂, Vm̂), n̂, m̂⟩ : x̂ ∈ O(x, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}
= {⟨φ_{β}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}

by the assumption that α = β + 1 is the first place at which the lemma runs afoul; but this last statement amounts to asserting φ_{β+1}(x, U_{n}, V_{m}) = φ_{β+1}(y, U_{n}, V_{m}).

On the other hand if α is a limit and φ_{α+1}(x, U_{n}, V_{m}) = φ_{α+1}(y, U_{n}, V_{m}), then again by definition

{⟨φ_{α}(x̂, Un̂, Vm̂), n̂, m̂⟩ : x̂ ∈ O(x, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}
= {⟨φ_{α}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}},

and by the assumption that α a limit we have

{⟨φ_{β}(x̂, Un̂, Vm̂), β, n̂, m̂⟩ : x̂ ∈ O(x, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}, β < α}
= {⟨φ_{β}(ŷ, Un̂, Vm̂), β, n̂, m̂⟩ : ŷ ∈ O(y, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}, β < α}.

This implies in particular that

{⟨φ_{β}(x̂, U_{n}, V_{m}), β, n, m⟩ : x̂ ∈ O(x, U_{n}, V_{m}), β < α}
= {⟨φ_{β}(ŷ, U_{n}, V_{m}), β, n, m⟩ : ŷ ∈ O(y, U_{n}, V_{m}), β < α}.

Since for ẑ ∈ O(z, U, V) one always has φ_{β}(z, U, V) = φ_{β}(ẑ, U, V) this entails

{⟨φ_{β}(x, U_{n}, V_{m}), β, n, m⟩ : β < α} = {⟨φ_{β}(y, U_{n}, V_{m}), β, n, m⟩ : β < α},

as desired.

Corollary 6.26. For α < β,

φ_{β}(x, U_{n}, V_{m}) = φ_{β}(y, U_{n}, V_{m})

implies

φ_{α}(x, U_{n}, V_{m}) = φ_{α}(y, U_{n}, V_{m}).

Proof. By transfinite induction on β.

Lemma 6.27. φ_{x} = φ_{γ(x)+2}(y, X, G) implies γ(x) = γ(y) and therefore

φ_{x} = φ_{y}.

Proof. Claim(I): For all y_{0}, y_{1} ∈ [y]_{G}, n, m ∈ ℕ

φ_{γ(x)}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)}(y_{1}, U_{n}, V_{m})

implies

φ_{γ(x)+1}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)+1}(y_{1}, U_{n}, V_{m}).

Proof of claim: If not, then from

{⟨φ_{γ(x)+1}(y′, U_{n}, V_{m}), n, m⟩ : n, m ∈ ℕ, y′ ∈ [y]_{G}}

equaling

{⟨φ_{γ(x)+1}(x_{0}, U_{n}, V_{m}), n, m⟩ : n, m ∈ ℕ, x_{0} ∈ [x]_{G}}

we can obtain x_{0}, x_{1} ∈ [x]_{G} so that

φ_{γ(x)+1}(x_{1}, U_{n}, V_{m}) = φ_{γ(x)+1}(y_{1}, U_{n}, V_{m}) ≠
φ_{γ(x)+1}(x_{0}, U_{n}, V_{m}) = φ_{γ(x)+1}(y_{0}, U_{n}, V_{m}).

By 6.25

φ_{γ(x)}(x_{1}, U_{n}, V_{m}) = φ_{γ(x)}(y_{1}, U_{n}, V_{m}) =
φ_{γ(x)}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)}(x_{0}, U_{n}, V_{m}),

with a contradiction to the definition of γ(x). (Claim(I)□) Claim(II): For all β < ω_{1}, y_{0}, y_{1} ∈ [y]_{G}, n, m ∈ ℕ

φ_{γ(x)}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)}(y_{1}, U_{n}, V_{m}) ⇒ φ_{β}(y_{0}, U_{n}, V_{m}) = φ_{β}(y_{1}, U_{n}, V_{m}).

Proof of claim: By induction on β ≤ γ(x) when it is trivial, as is carrying the induction through limit stages. So let us suppose the claim is true at β and attempt to lift up to β + 1. For this purpose suppose y_{0}, y_{1} ∈ [y]_{G}, n, m ∈ ℕ

φ_{γ(x)}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)}(y_{1}, U_{n}, V_{m}).

Then by claim(I) we have

φ_{γ(x)+1}(y_{0}, U_{n}, V_{m}) = φ_{γ(x)+1}(y_{1}, U_{n}, V_{m}),

and so by definition

{⟨φ_{γ(x)}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y_{0}, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}} =
{⟨φ_{γ(x)}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y_{1}, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}},

and so by the inductive assumption

{⟨φ_{β}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y_{1}, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}
= {⟨φ_{β}(ŷ, Un̂, Vm̂), n̂, m̂⟩ : ŷ ∈ O(y_{0}, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}
∴ φ_{β+1}(y_{0}, U_{n}, V_{m}) = φ_{β+1}(y_{1}, U_{n}, V_{m}).
(Claim(II)□)
Hence γ(y) ≤ γ(x).

For the reverse direction note then that φ_{y} = φ_{γ(y)+2}(x, X, G) and apply the already proved part of this lemma.

The next two lemmas are technical in appearance but nearly vacuous in content. The first states that in some sense φ_{α}(x, U, V) provides information about x that is in some sense isomorphic to the information about g · x provided by

φ_{α}(g · x, g · U, gV g^{−1});

this should hardly be surprising, since we can imagine an isomorphism of the whole complex (X, G) which respects the structure provided by the group action and is given by

x ↦ g · x,
h ↦ ghg^{−1}.

There is a minor issue here that the choice of the bases {U_{n}} and {V_{m}} may be affected by translation and conjugation by g, and so for strictly notational convenience the lemma is stated just for g ∈ G_{0}. The second lemma states that as we restrict U and V the φ_{α}(x, U, V) invariant provides more information. Again this has an intuitive basis, since φ_{α}(x, U, V) tries to sample information about x by restricting its movement to the local orbit O(x, U, V); as we restrict the local orbit further the likelihood of confusing x with some other point diminishes.

Lemma 6.28. For g ∈ G_{0}, the countable dense subgroup fixed at 6.21, U ⊂ X, V ⊂ G open, and x, y ∈ U, 1 ∈ V, V^{−1} = V, α < ω_{1} we have

φ_{α}(x, U, V) = φ_{α}(y, U, V)

implies

φ_{α}(g · x, g · U, gV g^{−1}) = φ_{α}(g · y, g · U, gV g^{−1}).

Proof. At the case α = 0 it is immediate, since for any l, l̂ with g · U_{l} = Ul̂

l ∈ φ_{0}(x, U, V) ⇔ l̂∈ φ_{0}(g · x, g · U, gV g^{−1}).

The rest proceeds by transfinite induction, with the limit case trivial and the successor case resting on

φ_{α+1}(g · x, g · U, gV g^{−1}) =

{⟨φ_{α}(x̂, Un̂, Vm̂), n̂, m̂⟩ : x̂ ∈ O(g · x, g · U, gV g^{−1}), Un̂ ⊂ g · U, Vm̂ ⊂ gV g^{−1}},

which in turn equals the set of ⟨φ_{α}(g · x̄, g · U_{n}, gV_{m}g^{−1}), n̂, m̂⟩ such that

x̄ ∈ O(x, U, V),
U_{n} ⊂ U,
V_{m} ⊂ V,
Un̂ = g · U_{n},
Vm̂ = gV_{m}g^{−1}.

by the assumption that {U_{n} : n ∈ ℕ} and {V_{m} : m ∈ ℕ} are invariant sets under translations and conjugation by elements of G_{0}.

Lemma 6.29. For x, y ∈ Û ⊂ U, 1 ∈ V̂ ⊂ V, V = V^{−1}, V̂ = V̂^{−1}, α < ω_{1} we have

φ_{α}(x, Û, V̂) = φ_{α}(y, Û, V̂)

implies

φ_{α}(x, U, V) = φ_{α}(y, U, V).

Proof. For the case α = 0 note that the set of

O_{l,U,V} =_{df} {z ∈ U : l ∈ φ_{0}(z, U, V)}

is an open set, and l ∈ φ_{0}(z, U, V) if and only if O(z, Û, V̂) ∩ O_{l,U,V} ≠ ∅.

The lemma is then proved by transfinite induction. The limit steps are trivial. For the successor step, from assuming inductively that the lemma is true at α,

φ_{α+1}(x, Û, V̂) = φ_{α+1}(y, Û, V̂),

and Û ⊂ U, V̂ ⊂ V it follows that

{φ_{α}(x̂, U_{n}, V_{m}) : U_{n} ⊂ Û, V_{m} ⊂ V̂ , x̂ ∈ O(x, Û, V̂)}
= {φ_{α}(ŷ, U_{n}, V_{m}) : U_{n} ⊂ Û, V_{m} ⊂ V̂ , ŷ ∈ O(y, Û, V̂)},

and hence

{φ_{α}(x̂, U_{n}, V_{m}) : U_{n} ⊂ U, V_{m} ⊂ V, x̂ ∈ O(x, U, V)}
= {φ_{α}(ŷ, U_{n}, V_{m}) : U_{n} ⊂ U, V_{m} ⊂ V, ŷ ∈ O(y, U, V)},

by the inductive assumption, 6.28, and the observation that for each x̂ ∈ O(x, U, V), U_{n} ⊂ U containing x̂, and V_{m} ⊂ V we can find g ∈ G_{0}, n̂, m̂ ∈ ℕ with

g · x̂ ∈ O(x, Û, V̂),
g · x̂ ∈ Un̂ ⊂ Û ∩ g · U_{n},
Vm̂ ⊂ V̂ ∩ gV_{m}g^{−1}.

Lemma 6.30. X, G, and so on, as above. There is a countable language L and a a Borel map θ : X → Mod(L) such that:

(i) xE_{G}y ⇒ θ(x) ≅ θ(y); (ii) θ(x) ≅ θ(y) ⇒ φ_{x} = φ_{y}; (iii) [θ[X]]_{S_{∞}} =_{df} {M ∈ Mod(L) : ∃x ∈ X(M ≅ θ(x))} is G_{δ}; (iv) moreover there is a Borel map

ρ : [θ[X]]_{S_{∞}} → X

such that for all M ∈ [θ[X]]_{S_{∞}}

θ(ρ(M)) ≅ M.

Proof. Let L = ⟨(O_{l})_{l∈ℕ}, (R_{n,m})_{n,m∈ℕ}, E, (F_{h})_{h∈G_{0}}⟩, where each O_{l} is a unary predicate, each R_{n,m} is a unary predicate, E is a binary relation, and each F_{h} is a unary function. Fix x ∈ X. We will go ahead and describe a model M(x). It is most convenient to first present it as a countable model whose underlying set is the union over m and n of the equivalence classes in G of the open equivalence relations ≈_{m,n}^{x}, as it was defined at 6.22; after the fact it will be easy to see that this can be reorganized with θ(x) ≅ M(x) a structure on ℕ with

x ↦ θ(x)

Borel.

As to motivation, the models M(x) can be thought of as being almost formed out of the local orbits for [x]_{G}. The O_{l} are intended to describe which open sets U_{l} intersect the local orbit, while the R_{n,m} indicate the U_{n} and V_{m} that define this “almost local orbit”. The F_{g} functions are included so as to enforce that M(x) has a kind of homogeneity, in turn used to obtain ρ as a Borel function. The E is not an equivalence relation, but corresponds to something like the membership relation, reflecting when φ_{α+1}(x, U_{n}, V_{m}) has ⟨φ_{α}(x̂, Un̂, Vm̂), n̂, m̂⟩ as an element for suitable n̂, m̂, x̂. Anyway, this is all only motivation and not yet an accurate description of the proof.

For h ∈ G and m, n ∈ ℕ with h · x ∈ U_{n}, we let [h]_{n,m}^{x} be the set of ĥ ∈ G for which there exists k ∈ ℕ, and

g_{0}, g_{1}, ..., g_{k−1} ∈ V_{m},

and for

h_{0} = h, h_{i+1} = g_{i}h_{i}

we have

h_{k} = ĥ

and each

h_{i} · x ∈ U_{n}.

So [h]_{n,m}^{x} is similar but not necessarily identical to the pullback to G of O(h · x, U_{n}, V_{m}) under the map g ↦ g · x; it is not identical in the case that the stabilizer of x is non-trivial – for if g · x = x but g not in V_{m} then

g · x ∈ O(x, U_{n}, V_{m})

without necessarily g ∈ [1_{G}]_{n,m}^{x}.

Define M(x) to be the set {⟨[h]_{n,m}^{x}, n, m⟩ : h ∈ G, m, n ∈ ℕ, h · x ∈ U_{n}}. Since any open equivalence relation on an open subset of G can have only countably many classes, each {⟨[h]_{n,m}^{x}, n, m⟩ : h ∈ G, h · x ∈ U_{n}} is countable, and so M(x) is countable.

We specify the behavior of the relations in M(x) as follows:-

(A) O_{l}: M(x) ⊧ O_{l}(⟨[h]_{n,m}^{x}, n, m⟩) if (and only if) l ∈ φ_{0}(h · x, m, n); (B) E: M(x) ⊧ ⟨[h]_{n,m}^{x}, n, m⟩E⟨[ĥ]_{n̂,m̂}^{x}, n̂, m̂⟩ if

U_{n} ⊂ Un̂, V_{m} ⊂ Vm̂, h ∈ [ĥ]_{n̂,m̂}^{x};

(C) R_{n,m}: M(x) ⊧ R_{n,m}(⟨[h]_{n̂,m̂}^{x}, n̂, m̂⟩) if m = m̂ and n = n̂;

(D) (F_{g}(⟨[h]_{n,m}^{x}, n, m⟩))^{M(x)} = ⟨[ĥ]_{n̂,m̂}^{x}, n̂, m̂⟩ if

g · U_{n} = Un̂,
gV_{m}g^{−1} = Vm̂,
gh ∈ [ĥ]_{n̂,m̂}^{x}.

Claim(I): If g ∈ G, x, y ∈ X with g · x = y, then there exists an isomorphism

π_{g} : M(x) ≅ M(y).

Proof of Claim: For any h ∈ G and n, m with h · x ∈ U_{n} let

π_{g}(⟨[h]_{n,m}^{x}, n, m⟩) = ⟨[hg^{−1}]_{n,m}^{y}, n, m⟩.

From the definition of the ≈_{n,m}^{x} equivalence relations this is well defined. It is immediate that the map respects the predicates of the form R_{m̂,n̂}. Since

O(h · x, U_{n}, V_{m}) = O(ĥ · y, U_{n}, V_{m}) = O(ĥg · x, U_{n}, V_{m}),

whenever π_{g}(⟨[h]_{n,m}^{x}, n, m⟩) = ⟨[ĥ]_{n,m}^{y}, n, m⟩, we have preservation of the O_{l} predicates and the binary relation E. Finally for the functions of the form Fĝ, ĝ ∈ G_{0}, given

(Fĝ(⟨[h]_{n,m}^{x}, n, m⟩))^{M(x)} = ⟨[ĝh]_{n̂,m̂}^{x}, n̂, m̂⟩,
π_{g}(⟨[h]_{n,m}^{x}, n, m⟩) = ⟨[hg^{−1}]_{n,m}^{y}, n, m⟩,

it follows

π_{g}(⟨[ĝh]_{n̂,m̂}^{x}, n̂, m̂⟩) = ⟨[ĝhg^{−1}]_{n̂,m̂}^{y}, n̂, m̂⟩
= (Fĝ(⟨[hg^{−1}]_{n̂,m̂}^{y}, n̂, m̂⟩))^{M(y)} = (Fĝ(π_{g}(⟨[h]_{n̂,m̂}^{y}, n̂, m̂⟩))^{M(y)}.
(Claim(I)□)

Claim(II): Let x, y ∈ X with

π : M(x) ≅ M(y)

an isomorphism. For all α < ω_{1}, if

π(⟨[h]_{n,m}^{x}, n, m⟩) = ⟨[ĥ]_{n,m}^{y}, n, m⟩

then

φ_{α}(h · x, n, m) = φ_{α}(ĥ · y, n, m).

Proof of Claim: By transfinite induction on α, with the base case α = 0 following from the fact that π respects the O_{l} predicates and the limit step being being trivial. For the successor step suppose the claim holds at α for all x, y and π, and then fix x, y ∈ X and

π : M(x) ≅ M(y)

with

π(⟨[h]_{n,m}^{x}, n, m⟩) = ⟨[ĥ]_{n,m}^{y}, n, m⟩.

By symmetry it suffices to show that

φ_{α+1}(h · x, U_{n}, V_{m})
=_{df} {⟨φ_{α}(x̂, Un̂, Vm̂), n̂, m̂⟩ : x̂ ∈ O(h · x, U_{n}, V_{m}), Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}}

is a subset of φ_{α+1}(ĥ · y, U_{n}, V_{m}). But for each

x̂ ∈ O(h · x, U_{n}, V_{m})

and

Un̂ ⊂ U_{n}, Vm̂ ⊂ V_{m}

we may find some g ∈ [h]_{n,m}^{x} with g · x = x̂. Fixing this g note that

M(x) ⊧ ⟨[g]_{n̂,m̂}^{x}, n̂, m̂⟩E⟨[h]_{n,m}^{x}, n, m⟩

and hence there will be some ĝ ∈ G with

M(y) ⊧ ⟨[ĝ]_{n̂,m̂}^{y}, n̂, m̂⟩E⟨[ĥ]_{n,m}^{y}, n, m⟩

and

π(⟨[g]_{n̂,m̂}^{x}, n̂, m̂⟩) = ⟨[ĝ]_{n̂,m̂}^{y}, n̂, m̂⟩.

From the definitions

⟨φ_{α}(ĝ · y, Un̂, Vm̂), n̂, m̂⟩ ∈ φ_{α+1}(ĥ · y, U_{n}, V_{m})

while by inductive assumption

φ_{α}(ĝ · y, Un̂, Vm̂) = φ_{α}(g · x, Un̂, Vm̂),

as required. (Claim(II) □)

It is now routine to find a Borel function θ : X →Mod(L) so that θ(x) ≅ M(x) for all x ∈ X. To see this note that M(x) has underlying set

{[g]_{n,m}^{x} : g ∈ G_{0}, n, m ∈ ℕ},

since G_{0} is dense and each ≈_{n,m}^{x} is an open equivalence relation. Recalling our fixed enumeration (g_{i})_{i∈ℕ} of G_{0}, the function

X → 2^{ℕ}
x ↦ A_{x} = {s(i, n, m) ∈ ℕ : g_{i} · x ∈ U_{n}, ∀j < i(g_{i} ≉_{n,m}^{x} g_{j})}

is Borel, where s(·, ·, ...) : ℕ^{<ℕ} → ℕ is some long previously chosen bijection between ℕ^{<ℕ} and ℕ. We can now define θ(x) ∈ Mod(L) by: θ(x) ⊧ O_{l}(k) if and only for s(i, n, m) the kth element of A_{x} we have

l ∈ φ_{0}(g_{i} · x, n, m);

θ(x) ⊧ R_{n,m}(k) if and only for s(i, n̂, m̂) the kth element of A_{x} we have n = n̂, m = m̂; and so on. Since functions such as

X × ℕ × ℕ × ℕ → 2^{ℕ}
(x, i, n, m) ↦ φ_{0}(g_{i} · x, U_{n}, V_{m})

are Borel it is not hard to see that the resulting map θ : X → Mod(L) is Borel.

Thus we have our Borel map and by claims (I) and (II) we have (i) and (ii) from the conclusion of the lemma. There is still a debt for (iii), to the effect that the range of θ has Borel saturation, and for (iv), to the effect that there exists an orbit inverse. In order to honor this let Y be the set of all M ∈ Mod(L) for which (1)-(7) below hold:

(1) the R_{n,m}’s uniquely label: for all a ∈ ℕ there are unique n, m ∈ ℕ for which M ⊧ R_{n,m}(a); for all a ∈ ℕ, n, m ∈ ℕ, if M ⊧ R_{n,m}(a) then U_{n} ≠ ∅, and moreover if U_{n} ⊂ Un̂, V_{m} ⊂ Vm̂ then there is a unique b ∈ ℕ with

M ⊧ R_{n̂,m̂}(b), aEb;

if

M ⊧ aEb, R_{n,m}(a), R_{n̂,m̂}(b)

then U_{n} ⊂ Un̂, V_{m} ⊂ Vm̂;

(2) the O_{l} predicates remember the structure held by {U_{l} : l ∈ ℕ}: for all a ∈ ℕ, n, m, l ∈ ℕ, m̂ ∈ ℕ with Vm̂ ⊂ V_{m}, if M ⊧ O_{l}(a), R_{n,m}(a) then there is some b ∈ ℕ,

M ⊧ bEa, R_{l,m̂}(b);

moreover M ⊧ O_{l}(a) implies that for each k ∈ ℕ there exists l̂with Ul̂ ≠ ∅,

d(Ul̂) < 2^{−k}

(d the complete metric on X),

\overline{Ul̂} ⊂ U_{l}

and

M ⊧ Ol̂(a);

(3) in pursuit of the idea that F_{g}(a) = Fĝ(a) exactly when some open set U_{l} with O_{l}(a) requires it: for all a, b ∈ ℕ, n, m, ∈ ℕ, g, ĝ ∈ G_{0} with M ⊧ aEb, R_{n,m}(b) we have

M ⊧ F_{g}(b) = Fĝ(b)

if and only if there is some some l ∈ ℕ such that

M ⊧ O_{l}(a)

and there are

h_{0}, h_{1}, ..., h_{k} ∈ V_{m} ∩ G_{0}

so that each

h_{i}h_{i−1}...h_{0} · U_{l} ⊂ U_{n}

and

ĝh_{k}h_{k−1}....h_{0} = g;

(4) when aEb, the collection of O_{l}’s met by a determine the signature for b: for all a, b ∈ ℕ, n, m, l ∈ ℕ such that

M ⊧ aEb, R_{n,m}(b)

we have

M ⊧ O_{l}(b)

if and only if there exists l̂∈ ℕ and h_{0}, h_{1}, ..., h_{k} ∈ V_{m} ∩ G_{0} so that each

h_{i}h_{i−1}...h_{0} · Ul̂ ⊂ U_{n},
h_{k}h_{k−1}...h_{0} · Ul̂ ⊂ U_{l}

and

M ⊧ Ol̂(a);

(5) the E relation provides a partial order respected by the F_{g} functions that in turn enforce a type of homogeneity: for all a, b ∈ ℕ, g ∈ G_{0}

M ⊧ aEb ⇒ F_{g}(a)EF_{g}(b);

for all a, b ∈ ℕ there exists g ∈ G_{0}, c ∈ ℕ such that

M ⊧ cEa, F_{g}(c)Eb;

and

M ⊧ ∀a, b, c(aEb ∧ bEc ⇒ aEc);
M ⊧ ∀a, b((aEb ∧ bEa) ⇒ a = b);

(6) the composition of the functions respects the group operations....:(F_{1_{G}})^{M} is the identity; for g, ĝ ∈ G,

(F_{g} ∘ Fĝ)^{M} = (F_{gĝ})^{M};

...and the functions themselves the open sets: for a ∈ ℕ g ∈ G_{0}, l, l̂ ∈ ℕ, g · U_{l} = Ul̂ implies

M ⊧ O_{l}(a) ⇔ Ol̂(F_{g}(a)),

and if n, m, n̂, m̂ ∈ ℕ with g · U_{n} = Un̂, gV_{n}g^{−1} = Vm̂ then

M ⊧ R_{n,m}(a) ⇔ R_{n̂,m̂}(F_{g}(a));

(7) the E structure of the (F_{g})_{g∈G_{0}}-closure of a point is determined by its open sets: for all g, ĝ ∈ G_{0}, a, b, c ∈ ℕ, n, m, n̂, m̂ ∈ ℕ with

M ⊧ aEb, aEc, R_{n,m}(b), R_{n̂,m̂}(c)

we have that

M ⊧ F_{g}(b)EFĝ(c)

if and only if

g · U_{n} ⊂ ĝ · Un̂,
gV_{m}g^{−1} ⊂ ĝVm̂ĝ^{−1},

and there exists l ∈ ℕ, h_{0}, ..., h_{k} ∈ G ∩ Vm̂ such that

M ⊧ O_{l}(a),

each h_{j}h_{j−1}...h_{0} · U_{l} ⊂ Un̂, and ĝh_{k}h_{k−1}....h_{0} = g.

Y is G_{δ} subset of Mod(L) since every condition is defined by a countable intersection of Boolean combinations of open sets. It is not hard to check that Y ⊂ [θ[X]]_{S_{∞}}; perhaps the most problematical of these conditions is (3), but that follows from the set {x : g_{i} ≈_{n,m}^{x} g_{j}} being open for each i, j, n, m.

We will find for each M ∈ Y some ρ(M) ∈ X such that θ(ρ(M)) ≅ M. It will be clear from the construction that ρ is Borel, and then (iii) and (iv).

Fixing M ∈ Y we build a sequence of a_{i} ∈ M, n(i), m(i) ∈ ℕ, such that for all i, b, l ∈ ℕ, g, ĝ ∈ G_{0}:

(a)_{i} M ⊧ a_{i+1}Ea_{i}; (b)_{i} M ⊧ R_{n(i),m(i)}(a_{i}); (c)_{i} the closure of U_{n(i+1)} is included in U_{n(i)}; V_{m(i+1)} ⊂ V_{i} ∩ V_{m(i)}; (d)_{i} d(U_{n(i)}) < 2^{−i}; U_{n(i)} ≠ ∅; (e)_{b} there exists some g ∈ G_{0} and j ∈ ℕ with

M ⊧ F_{g}(a_{n(j)})Eb;

(f)_{g,ĝ,i} if

M ⊧ F_{g}(a_{i}) = Fĝ(a_{i})

then there exists some j > i and h_{0}, h_{1}, ..., h_{k} ∈ V_{m(i)} such that for each p ≤ k

h_{p}h_{p−1}...h_{0} · U_{n(j)} ⊂ U_{n(i)},

and

gh_{k}h_{k−1}...h_{0} = ĝ;

(g)_{i,l} if

M ⊧ O_{l}(a_{i})

then there is some j > i and

h_{0}, h_{1}, ..., h_{k} ∈ V_{m}(i) ∩ G_{0}

so that each

h_{i}h_{i−1}...h_{0} · U_{n(j)} ⊂ U_{n(i)},

and

h_{k}h_{k−1}...h_{0} · U_{n(j)} ⊂ U_{l};

The main issue is completing (f) and (e), for which we fix some onto

f : ℕ ↠ ℕ × ℕ ∪ G_{0} × G_{0} × ℕ

which will remind us to eventually serve each requirement of the form (e)_{b,l}, (f)_{g,ĝ,i}, or (g)_{i,l}. Assume that f is not only onto but the preimage of each point under f is infinite, (so for (f)_{g,ĝ,j} and (g)_{j,l} we will be prompted at some later i > j). Given that we have constructed a_{i}, n(i), m(i), we use (2) to choose some a ∈ ℕ and n̂, m̂ ∈ ℕ with with d(Un̂) < 2^{−i−1}, \overline{Un̂} ⊂ U_{n(i)},

M ⊧ aEa_{i}, R_{n̂,m̂}(a);

if f(i) = (2b, l) some b ∈ ℕ then we find a_{i+1}Ea and g ∈ G_{0} with

M ⊧ F_{g}(a_{i+1})Eb

by (5); if f(i) = (g, ĝ, j), j < i, then we can find a_{i+1}Ea suitable for (f)_{g,ĝ,i} by (3); if f(i) = (2j + 1, l), j < i then we find the required a_{i+1} that fulfills (g)_{i,l} using (4) and (2).

Supposing now that all this is put into place, then let x ∈ X be defined by

{x} = ⋂_{ℕ} U_{n(i)};

by (d) this x exists. Note that for each l we have x ∈ U_{l} if and only if there is some i with

\overline{U_{n(i)}} ⊂ U_{l}.

Note also by (5) that for each i < j we have

M ⊧ a_{j}Ea_{i}.

Claim(III): For each i, n̂, m̂, l̂∈ ℕ, g ∈ G_{0} with g · U_{n(i)} = Un̂ and gV_{m(i)}g^{−1} = Vm̂

l̂∈ φ_{0}(g · x, Un̂, Vm̂) ⇔ M ⊧ Ol̂(F_{g}(a_{i})).

Proof of Claim: First note that for the case g = 1_{G}, l ∈ φ_{0}(x, U_{n(i)}, V_{m(i)}) if and only if

M ⊧ O_{l}(a_{i}),

since the ‘if’ direction holds by (g)_{i,l} and the ‘only if’ direction follows from (4). Now the claim follows by (6). (Claim(III)□)

Now by (4) again it follows that if b ∈ ℕ, n, m, l, i ∈ ℕ,

M ⊧ a_{i}Eb ∧ R_{n,m}(b)

then

l ∈ φ_{0}(x, U_{n}, V_{m}) ⇔ M ⊧ O_{l}(b);

then more generally by (6) again we have that for all if b ∈ ℕ, n, m, l, i ∈ ℕ, g ∈ G_{0} such that

M ⊧ F_{g}(a_{i})Eb, R_{n,m}(b)
l ∈ φ_{0}(g · x, U_{n}, V_{m}) ⇔ M ⊧ O_{l}(b).

Thus we obtain a map π that respects all the O_{l} predicates if we go right ahead and define

π : M(x) → M

by the formulation that

π(⟨[g]_{n,m}^{x}, n, m⟩) = b

if and only if for some i ∈ ℕ

M ⊧ R_{n,m}(b) ∧ F_{g}(a_{i})Eb.

From (1) and (3) this is well defined and everywhere defined. It is one to one by (f) and onto by (e) and (1). From (3) and (6) we have that π respects the functions of the form F_{g}, in the sense that

(F_{g})^{M}(π(d)) = π((F_{g})^{M(x)}(d))

for all d ∈ M(x). Finally for the E relation we use (7).

Definition 6.31. For X, G and so on as above, and x ∈ X, let X(φ_{x}) be the set

{y ∈ X : φ_{y} = φ_{x}}

which equals

{y ∈ X : φ_{γ(x)+2}(y, X, G) = φ_{x}}

by 6.27. Let τ(φ_{x}) be the topology on X(φ_{x}) with basic open sets of the form

U(φ_{γ(x)}(x̂, n, m), n, m) =
{y ∈ X(φ_{x}) ∩ U_{n} : φ_{γ(x)}(y, U_{n}, V_{m}) = φ_{γ(x)}(x̂, U_{n}, V_{m})}

for x̂ ∈ X.

This is a basis by 6.29.

Lemma 6.32. For x ∈ X as above, (X(φ_{x}), τ(φ_{x})) is a Polish space.

Proof. We produce a sequence of Polish topologies on X,

(τ_{i,j,α})_{i∈ℕ,j∈{0,1,2},α≤γ(x)+2},

so that for i ∈ ℕ, j ∈ {0, 1, 2}, α < β we have τ_{i,j,α} ⊂ τ_{i′,j′,α′}, whenever (i, j, α) < (i′, j′, α′) in the reverse lexicographic ordering (which is to say, α < α′, or α = α′, j < j′, or α = α′, j = j′, i < i′). τ_{i,j,α} restricted to X(φ_{x}) will be included in τ(φ_{x}). At the very last stage we will have that X(φ_{x}) is an open subset of (X, τ_{0,2,γ(x)+2}).

(0) τ_{0,0,0} is the original topology on X.

(1) Each τ_{i+1,j,α} will have a basis of the form

B_{i+1,j,α} = {C ∩ W ∩ O : O ∈ B_{i,j,α}} ⋃ B_{i,j,α}

where B_{i,j,α} is a basis for τ_{i,j,α} and C is some τ_{i,j,α}-closed set and W is some τ_{i,j,α}-open set. From this it follows inductively that if (X, τ_{i,j,α}) is Polish then by 2.3 so too (X, τ_{i+1,j,α}).

(2) At the limit steps of this construction, we let τ_{0,0,α} be the topology generated by the union of {τ_{i,j′,β} : (i, j) ∈ ℕ × {0, 1, 2}, β < α}, τ_{0,1,α} the topology generated by {τ_{i,0,α} : i ∈ ℕ}, τ_{0,2,α} the topology generated by {τ_{i,1,α} : i ∈ ℕ}, all of which will be Polish again by 2.3 and have as a basis ⋃{B_{i,j,β} : i ∈ ℕ, j ∈ {0, 1, 2}, β < α}, ⋃{B_{i,0,α} : i ∈ ℕ}, ⋃{B_{i,1,α} : i ∈ ℕ} respectively.

(3) Finally we will have that for each n, m, x̂ ∈ [x]_{G} ∩ U_{n} that

{y ∈ X : φ_{α}(y, U_{n}, V_{m}) = φ_{α}(x̂, U_{n}, V_{m})} ∪ (X \ U_{n})

is closed in τ_{0,1,α},

{y ∈ U_{n} : φ_{α}(y, U_{n}, V_{m}) = φ_{α}(x̂, U_{n}, V_{m})}

open in τ_{0,2,α}, and that

{y ∈ U_{n} : φ_{α}(y, U_{n}, V_{m}) ∉ {φ_{α}(g_{i} · x, U_{n}, V_{m}) : g_{j} · x ∈ U_{n}}}

is open in τ_{0,0,α+1}.

Granted that this may be done then we are certainly finished, since

{y ∈ X : φ_{γ(x)+2}(y, X, G) = φ_{γ(x)+2}(x, X, G)}

will be τ_{0,2,γ(x)+2}-open by (3).

The main part of this construction then is clearly in defining the closed sets to be added at each stage. The precise enumeration is given in the next paragraphs, but the idea is that B_{0,1,0} should have every (formerly closed intersection open) set of the form

{y ∈ g_{j}^{−1} · U_{n} : l ∉ φ_{0}(g_{j} · y, U_{n}, V_{m})},

B_{0,2,0} should have all sets of the form

{y ∈ g_{j}^{−1} · U_{n} : φ_{0}(g_{j} · y, U_{n}, V_{m}) = φ_{0}(g_{i} · x, U_{n}, V_{m})},

and B_{0,0,1} should have all sets of the form

{y ∈ g_{j}^{−1} · U_{n} : φ_{0}(g_{j} · y, U_{n}, V_{m}) ∉ {φ_{0}(g_{i} · x, U_{n}, V_{m}) : g_{i} · x ∈ U_{n}}}.

Similarly B_{0,1,α} will have every set of the form

⟨φ_{β}(g_{i} · x, Un̂, Vm̂), n̂, m̂⟩ ∉ φ_{α}(g_{j} · y, U_{n}, V_{m})

for β < α. B_{0,2,α} will have every set of the form

φ_{α}(g_{i} · x, U_{n}, V_{m}) = φ_{α}(g_{j} · y, U_{n}, V_{m}),

and B_{0,0,α+1} will have every set of the form

φ_{α}(g_{j} · y, U_{n}, V_{m}) ∉ {φ_{α}(g_{i} · x, U_{n}, V_{m}) : g_{i} · x ∈ U_{n}}.
Choose bijections s_{4} : ℕ^{4} → ℕ and s_{3} : ℕ^{3} → ℕ. Begin by defining
C_{s_{4}(l,n,m,j),0} = {y ∈ U_{n} : l ∉ φ_{0}(g_{j} · y, U_{n}, V_{m})} ⋃ X \ U_{n}

and

W_{s_{4}(l,n,m,j),0} = U_{n}.

So each C_{i,0} is closed, each W_{i,0} is open, and we may let τ_{0,0,0} be the original topology on X, B_{0,0,0} = {U_{l} : l ∈ ℕ}, and at each i

B_{i+1,0,0} = {U ∩ U_{i} ∩ C_{i} : U ∈ B_{i,0,0}} ∪ B_{i,0,0}

and take τ_{i+1,0,0} to be the topology generated by B_{i+1,0}, and when we move on to τ_{0,1,0} = ⋃_{i∈ℕ} τ_{i,0,0} we have in particular that each

{y ∈ g_{j}^{−1} · U_{n} : φ_{0}(g_{j} · y, U_{n}, V_{m}) = φ_{0}(g_{i} · x, U_{n}, V_{m})} ∪ (X \ U_{n})

is closed for i, j ∈ ℕ, g_{i} · x ∈ [x]_{G} ∩ U_{n}.

By the same device define τ_{s_{4}(i,n,m,j)+1,1,0} so that

{y ∈ g_{j}^{−1} · U_{n} : φ_{0}(g_{·}y, U_{n}, V_{m}) = φ_{0}(g_{i} · x, U_{n}, V_{m})}

becomes open, since each of these sets were previously relatively closed in g_{j}^{−1} · U_{n} in τ_{0,1,0}. We let τ_{0,2,0} be the union of the topologies of the form τ_{i,1,0}.

Then we define τ_{s_{3}(n,m,j)+1,2,0} so that the set

{y ∈ g_{j}^{−1} · U_{n} : φ_{0}(g_{j} · y, U_{n}, V_{m}) ∉ {φ_{0}(g_{i} · x, U_{n}, V_{m}) : g_{i} · x ∈ U_{n}}}

now becomes open. We let τ_{0,0,1} be the union of the topologies τ_{i,2,0}.

So we can simply continue ad infinitum, steadily adding in for each α and i the set

{y ∈ g_{j}^{−1} · U_{n} : φ_{α}(g_{j} · y, U_{n}, V_{m}) = φ_{α}(g_{i} · x, U_{n}, V_{m})}

first as a relatively closed set, and then as an open set, and then adding in each set of the form

{y ∈ g_{j}^{−1} · U_{n} : φ_{α+1}(g_{j} · y, U_{n}, V_{m}) ⊂ φ_{α+1}(x, X, G)}

as a relatively closed subset of g_{j}^{−1} · U_{n}, and then as an open set in the space. Then we obtain sets of the form

{y ∈ g_{j}^{−1} · U_{n} : φ_{α+1}(g_{j} · y, U_{n}, V_{m}) = φ_{α+1}(g_{i} · x, U_{n}, V_{m})}

as relatively closed subsets of the g_{j}^{−1}U_{n}, which we can then proceed to make open.

Note that this never causes us to exceed the topology τ(φ_{x}) on X(φ_{x}), since for α > γ(x)

{y ∈ U_{n} ∩ X(φ_{x}) : φ_{α}(y, U_{n}, V_{m}) = φ_{α}(g_{i} · x, U_{n}, V_{m})}

equals

{y ∈ U_{n} ∩ X(φ_{x}) : φ_{γ(x)}(y, U_{n}, V_{m}) = φ_{γ(x)}(g_{i} · x, U_{n}, V_{m})}.

In the induced action of G on X(φ_{x}) obtained by restricting the original action on X:

Lemma 6.33. (X(φ_{x}), τ(φ_{x})) is a Polish G-space.

Proof. By consideration of the basis. If y_{0} ∈ X(φ_{x}) and g_{0} ∈ G, g_{0} · y_{0} ∈ W, for W some τ(φ_{x})-open set, then for the purposes of checking that the action is continuous we may assume that

W = {z ∈ X(φ_{x}) : φ_{α}(z, U_{n}, V_{m}) = φ_{α}(g_{0} · y_{0}, U_{n}, V_{m})},

some U_{n} containing g_{0} · y_{0} and some m ∈ ℕ. We can therefore find a small open set U containing y_{0}, V containing g_{0} with V · U ⊂ U_{n}; and thus for any y ∈ g_{0}^{−1} · U_{n} with

φ_{α}(y, g_{0}^{−1} · U_{n}, g_{0}^{−1}V_{m}g_{0}) = φ_{α}(y_{0}, g_{0}^{−1} · U_{n}, g_{0}^{−1}V_{m}g_{0})

and for any g ∈ V ∩ V_{m}g_{0}, we have by 6.28

φ_{α}(g · y, U_{n}, V_{m}) = φ_{α}(g_{0} · y_{0}, U_{n}, V_{m}).

By 6.29, the set of y ∈ X(φ_{x}) with

φ_{α}(y, g_{0}^{−1} · U_{n}, g_{0}^{−1}V_{m}g_{0}) = φ_{α}(y_{0}, g_{0}^{−1} · U_{n}, g_{0}^{−1}V_{m}g_{0})

is open.

Lemma 6.34. Every orbit in (X(φ_{x}), τ(φ_{x})) is dense.

Proof. Since X(φ_{x}) is essentially defined to be the collection of elements of X that meet exactly the needed open sets.

Lemma 6.35. If every orbit in (X(φ_{x}), τ(φ_{x})) is meager then the action of G on this space is turbulent.

Proof. Since we then have that each y ∈ X(φ_{x}) ∩ U_{n} will have some x̂ ∈ [x]_{G} with

φ_{γ(x)}(y, U_{n}, V_{m}) = φ_{γ(x)}(x̂, U_{n}, V_{m}).

Then

O(y, U(φ_{γ(x)}(x̂, n, m)), V_{m})(= O(y, U_{n}, V_{m}))

is dense in U(φ_{γ(x)}(x̂, n, m)).

Definition 6.36. A Polish group H is a GE group if whenever Y is a Polish H-space and y ∈ Y has [y]_{H} a G_{δ}, then for all z ∈ Y

\overline{[y]_{H}} = \overline{[z]_{H}}

implies zE_{H}y.

In other words, a dense G_{δ} orbit under a GE group takes over the space, allowing no other orbits to share the precise same closure.

The class of GE groups is surprisingly broad. It includes locally compact Polish groups, since in this case every orbit must be F_{σ}. Less obviously it includes abelian and even nilpotent, as well as invariantly metrizable and ℵ_{0} products of locally compact.

The following lemma seems to be have been discovered by various people, including David Marker and Ramez Sami. Although I cannot be sure as to its first author, something similar is already implicit in [14].

Lemma 6.37. Let H be a Polish group and Y a Polish space. If y ∈ Y with [y]_{H} non-meager, then [y]_{H} is G_{δ}.

Proof. After possible pruning down to the space

\overline{[y]_{H}}

we may assume that the orbit of y is dense. Since [y]_{H} is certainly no worse than Σ ∼_{1}^{1}, it has the Baire property. Thus from it being non-meager, we may find an open set U ⊂ Y, U ≠ ∅, so that [y]_{H} is comeager in U, and then by the density of [y]_{H} it is comeager in Y. So fix A a dense G_{δ}, A ⊂ [y]_{H}. Then by 2.47 A^{∗H} =_{df} {y ∈ Y : ∀^{∗}h ∈ H(h·x ∈ A)} is a comeager set, and therefore in particular non-empty.

Note that A^{∗H} ⊂ [y]_{H}, and so by invariance of these two sets and the fact that H acts transitively on [y]_{H}, these are equal. By 6.9 A^{∗H} is a G_{δ}.

Definition 6.38. For E and F equivalence relations on Polish Y_{0} and Y_{1}, write

E ≤_{F B} F

to indicate that there is a faithful Borel reduction of E to F, so that for some Borel θ : Y_{0} → Y_{1} witnessing E ≤_{B} F we have that for each invariant Borel set A ⊂ Y_{0} the saturation of its image, [θ[A]]_{F}, is Borel.

Gao in [23] produces orbit equivalence relations E_{H_{1}}, E_{H_{2}} with E_{H_{1}} ≤_{B} E_{H_{2}} but not E_{H_{1}} ≤_{F B} E_{H_{2}}. The example is non-trivial.

Definition 6.39. Let H be a Polish group, Y_{0}, Y_{1} Polish H-spaces. A function π : Y_{0} → Y_{1} is an H-embedding if it is one to one and for all y ∈ Y_{0}, h ∈ H

π(g · y) = g · π(y).

The existence of a Borel H-embedding as above implies E_{H}^{Y_{0}} ≤_{B} E_{H}^{Y_{1}}.

Theorem 6.40. Let G be a GE group and X a Polish G-space. Then the following are equivalent:

(I) E_{G}^{X} admits classification by countable models;

(II) there is a countable language L with E_{G}^{X} ≤_{F B}≅ |_{Mod(L)};

(III) x ↦ φ_{x} provides a complete invariant for the orbit equivalence relation; (IV) there is no turbulent Polish G-space Y and continuous G-embedding from Y to X.

Proof. (II) implies (I) at once from the definitions, and (III) implies (II) by 6.30. (I) implies (IV) since 3.19 states that Y as in (IV) cannot admit classification by countable models. We are still to see (IV) implying (III) – and this is where the assumption on G is played.

Supposing that x ↦ φ_{x} is not a complete invariant, find orbit inequivalent x, y ∈ X such that φ_{x} = φ_{y}. By 6.35 we only need show every orbit in X(φ_{x}) is meager. But this follows, since if z ∈ X(φ_{x}) is G_{δ} then it is already dense by 6.34, so

xE_{G}zE_{G}y

by G being GE.

Exercise 6.41. Let G = c_{0} and X = ℝ^{ℕ} with the action by translation. Show that for all x

γ(x) = 0.

Exercise 6.42. Let G = S_{∞} and V_{n} = {g ∈ S_{∞} : ∀i < n(g(i) = i)} as at 6.1. Let X be a Polish S_{∞}-space.

Provide the following partial reconciliation of the notation from §6.1 and §6.2: If x, y ∈ U ⊂ X with V_{n} · x, V_{n} · y ⊂ U then for all α < ω_{1}

φ_{α}(x, V_{n}) = φ_{α}(y, V_{n})

implies φ_{α}(x, U, V_{n}) = φ_{α}(y, U, V_{n}).

6.3. A counterexample

Although it is open whether (I), (II), and (IV) of 6.40 imply one another for arbitrary Polish groups, in general they are not equivalent to (III).

Definition 6.43. Let M_{∞}(ℝ) be the group of measure preserving transformations of (ℝ, λ), with transformations agreeing a.e. identified, λ being Lebesgue measure.

M_{∞}(ℝ) is Polish. For instance it can be viewed as a closed subgroup of the unitary group on L^{2}(ℝ, λ). More directly if we fix {O_{i} : i ∈ ℕ} a basis of ℝ with each λ(O_{i}) ≤ 1 then as in 2.11 a complete metric is obtained by letting d_{ℝ,λ}(π_{0}, π_{1}) equal

∑ 2^{−i}[λ(π_{0}(O_{i})Δπ_{1}(O_{i})) + λ(π_{0}^{−1}(O_{i})Δπ_{1}^{−1}(O_{i}))].

Definition 6.44. Let X(λ, 1) equal the set of

(f_{n})_{n∈ℕ} = f⃗ ∈ (L^{2}(ℝ, λ))^{ℕ},

each f_{n} : ℝ → ℝ square integrable, such that:

(1) for each n ∈ ℕ,

∫_{ℝ} |f_{n}|λ(x) = 1;
(2) for n ≠ m ∈ ℕ
∫_{ℝ} |f_{n} · f_{m}|λ(x) = 0;

(3) for each n and a.e. x ∈ ℝ

f_{n}(x) ∈ {0, 1}.

L^{2}(ℝ, λ) is a Polish space with the metric

d(f, f′) = (∫ (f − f′)^{2}λ(x))^{1/2}

and the understanding that functions agreeing a.e. are identified. X(λ, 1) is a closed subspace of (L^{2}(ℝ, λ))^{ℕ}, and hence Polish. There is a natural action of M_{∞}(ℝ) on L^{2}(ℝ, λ): for π ∈ M_{∞}(ℝ), f ∈ L^{2}(ℝ, λ), x ∈ ℝ

(π · f)(x) = f(π^{−1}(x)).

This action is continuous and by taking its infinite product we obtain that (L^{2}(ℝ, λ))^{ℕ} is a Polish M_{∞}(ℝ)-space with the action

(π · f⃗)_{n}(x) = f_{n}(π^{−1}(x)).

X(λ, 1) is a closed invariant subspace of (L^{2}(ℝ, λ))^{ℕ}, and hence a Polish M_{∞}(ℝ) space.

Lemma 6.45. The orbit equivalence induced by the action of M_{∞}(ℝ) on X(λ, 1) is smooth: E_{M_{∞}(ℝ)}^{X(λ,1)} ≤_{B} id(ℝ).

Proof. A complete invariant for any f⃗ ∈ X(λ, 1) is given by the measure of {x ∈ ℝ : ∀n(f_{n}(x) = 0)}. Thus

θ : X(λ, 1) → ℝ
f⃗ ↦ 2^{−λ({x∈ℝ:∀n(f_{n}(x)=0)})}

provides the needed reduction.

Lemma 6.46. For all f⃗ ∈ X(λ, 1) and

U ⊂ X(λ, 1)

open containing f⃗,

V ⊂ M_{∞}(ℝ)

a neighborhood of the identity, there is open Û ⊂ U containing f⃗ such that for all h⃗ ∈ Û the set

O(h⃗, Û, V)

is dense in Û.

Proof. Fix f⃗, U, V as indicated, and choose δ > 0 small enough that for some N ∈ ℕ

Û =_{df} {g⃗ ∈ X(λ, 1) : ∀n < N(d(g_{n}, f_{n}) < δ)} ⊂ U,
B_{δ}(1) ⊂ V,

where B_{δ}(1) is the open d_{ℝ,λ} ball of radius δ about the identity of the group. Supposing g⃗, h⃗ ∈ Û and N ≤ M ∈ ℕ it suffices to find π · h⃗ ∈ O(f⃗, Û, V) such that (π · h⃗)_{n} = g_{n} for all n ≤ M.

For k⃗ ∈ X(λ, 1) and n ∈ ℕ let

A(n, k⃗) = {x ∈ ℝ : k_{n}(x) = 1}.

110 6. A GENERALIZED SCOTT ANALYSIS

Choose π_{M} ∈ M_{∞}(ℝ) such that

π_{M}^{2} = id,
π_{M}[A(M, h⃗) \ A(M, g⃗)] = A(M, g⃗) \ A(M, h⃗),

and for all x ∉ A(M, h⃗)Δ(M, g⃗)

π_{M}(x) = x.

Thus in particular we have for all k ∈ ℕ and x ∈ A(k, h⃗) ∩ A(k, g⃗)

π_{M}(x) = x. This can be repeated for h⃗^{1} = π_{M} · f, and we may successively obtain h⃗^{2}, h⃗^{3}, ..., h⃗^{M+1}

and

π_{M−1}, π_{M−2}, ..., π_{0}

such that at each n ≤ M

h⃗^{n+1} = π_{M−n} · h⃗^{n},
π_{M−n}^{2} = id,

π_{M−n}[A(M − n, h⃗^{n}) \ A(M − n, g⃗)] = A(M − n, g⃗) \ A(M − n, h⃗^{n}), and for all x ∉ A(M − n, h⃗^{n})Δ(M − n, g⃗)

π_{M−n}(x) = x.

Thus at each n ≤ M and k ∈ ℕ we have

A(k, h⃗^{n}) ∩ A(k, g⃗) ⊃ A(k, g⃗) ∩ A(k, h⃗),

and so each h⃗^{n} is in Û.

For each n we may partition A(M − n, h⃗^{n}) \ A(M − n, g⃗) into sets

B_{n,0}, B_{n,1}, ..., B_{n,k(n)}

with each λ(B_{n,i}) < δ^{2}/2, and then we may define π_{i,M−n} by

π_{i,M−n}|_{B_{n,i}} = π_{M−n}|_{B_{n,i}},

π_{i,M−n}|_{π[B_{n,i}]} = π_{M−n}|_{π_{i}[B_{n,i}]},
π_{i,M−n}|_{ℝ\(B_{n,i}∪π[B_{n,i}])} = id|_{ℝ\(B_{n,i}∪π[B_{n,i}])}.

Each such

π_{i,M−n} ∈ B_{δ}(1) ⊂ V.

Again for each k and n we have

π_{i,M−n}|_{A(k,h⃗^{n})∩A(k,g⃗)} = id|_{A(k,h⃗^{n})∩A(k,g⃗)}. Thus if let h⃗^{n,0} = h⃗^{n} and each h⃗^{n,i+1} = π_{i,M−n} · h⃗^{n,i} then we obtain each

h⃗^{n,i} ∈ O(h⃗, Û, V) and we finish. This suffices by the following completely general observation.

Lemma 6.47. Let G be a Polish group and X a Polish G-space in which every orbit is dense. Suppose as usual we fix some basis (U_{l})_{l∈ℕ} for X and a neighborhood basis (V_{n})_{n∈ℕ} around the identity in G. Suppose that for all n, m ∈ ℕ and x ∈ U_{n} there is some Û ⊂ U_{n} open containing x with every y ∈ Û having O(y, Û, V_{m}) dense in Û.

Then for all n ∈ ℕ, x, y ∈ U_{n}, and α < ω_{1}

φ_{α}(x, U_{n}, V_{m}) = φ_{α}(y, U_{n}, V_{m}).

Proof. For each m ∈ ℕ choose a basis {W_{n}^{m} : n ∈ ℕ} for X such that for each y ∈ W_{n}^{m} the local orbit O(y, W_{n}^{m}, V_{m}) is dense in W_{n}^{m}. By 6.29 it suffices to show that for all α < ω_{1}, n, m ∈ ℕ, x, y ∈ W_{n}^{m}

φ_{α}(x, W_{n}^{m}, V_{m}) = φ_{α}(y, W_{n}^{m}, V_{m}).

At once by the definition of the basis we have that for all n, m ∈ ℕ, x, y ∈ W_{n}^{m}

φ_{0}(x, W_{n}^{m}, V_{m}) = φ_{0}(y, W_{n}^{m}, V_{m}).

Now the general case follows by transfinite induction.

Thus by 6.27, 6.47, 6.46, for all f⃗, h⃗ ∈ X(λ, 1)
φf⃗ = φh⃗.

6.4. A different direction

Instead of measuring the extent to which the complete invariants for S_{∞} work for arbitrary Polish groups, we can also try to craft complete invariants for the general context. Here are just a few steps in this direction.

Ultimately we would want to show that the ≈_{γ} equivalence classes defined at 6.53 correspond to something comprehensible, such as in the case of S_{∞} when they can be thought of as hereditarily countable sets. At the very minimum one would hope to show that the ≈_{γ} equivalence relation is Borel reducible to an orbit equivalence relation arising from the group in question.

The approach here is abstract. It completes the task at 6.55, and obtains from a bound in the complexity of the individual orbits in X a bound in the Borel complexity of E_{G}^{X}, but it gives no further insight.

Definition 6.48. Let G be a Polish group and X a Polish G-space. For V_{0}, V_{1} ⊂ G open and non-empty, x_{0}, x_{1} ∈ X, write

(x_{0}, V_{0}) ≤_{1} (x_{1}, V_{1})

if

\overline{V_{0} · x_{0}} ⊂ \overline{V_{1} · x_{1}}.

Write

(x_{0}, V_{0}) ≤_{α+1} (x_{1}, V_{1})

if for all W_{0} ⊂ V_{0} open non-empty there is W_{1} ⊂ V_{1} non-empty and open with

(x_{1}, W_{1}) ≤_{α} (x_{0}, W_{0}).

For λ a limit

(x_{0}, V_{0}) ≤_{λ} (x_{1}, V_{1})

expresses that at every α < λ

(x_{0}, V_{0}) ≤_{α} (x_{1}, V_{1}).

Lemma 6.49. For W_{0} ⊂ V_{0}, W_{1} ⊃ V_{1}, all open and non-empty,

(x_{0}, V_{0}) ≤_{α} (x_{1}, V_{1})

implies

(x_{0}, W_{0}) ≤_{α} (x_{1}, W_{1}).

Proof. Immediate from the definitions.

From now on let B be a countable basis for G and let N be the non-empty elements of B. Assume G ∈ B.

Lemma 6.50.

(x_{0}, V_{0}) ≤_{α+1} (x_{1}, V_{1})

if and only if for all W_{0} ⊂ V_{0} in N there is W_{1} ⊂ V_{0} in N with

(x_{1}, W_{1}) ≤_{α} (x_{0}, W_{0}).

Proof. From 6.49.

Corollary 6.51. Let (V̂_{n})_{n∈ℕ} enumerate N. Then for all α < ω_{1}

{(x_{0}, x_{1}, n, m) : (x_{1}, V̂_{n}) ≤_{α} (x_{0}, V̂_{m})}

is a Π ∼_{α+k}^{0} set for some k ∈ ℕ.

Lemma 6.52. (a) Each ≤_{α} is transitive.

(b) α < β implies ≤_{α}⊃≤_{β} (in the sense that (x_{1}, W_{1}) ≤_{β} (x_{0}, W_{0}) implies (x_{1}, W_{1}) ≤_{α} (x_{0}, W_{0})).

Proof. (a) is almost immediate from the definitions. (b) routinely following by transfinite induction once we observe:

Claim: ≤_{1}⊃≤_{2}. Proof of claim: Suppose that (x_{1}, W_{1}) ≤_{2} (x_{0}, W_{0}). Choose U_{0} ⊂ X open

with W_{1} · x_{1} ∩ U_{0} ≠ ∅. Then we may choose some x′_{1} ∈ W_{1} · x_{1} ∩ U_{0} and U_{1} an open neighborhood of x′_{1} with \overline{U_{1}} ⊂ U_{0}. We then further choose some non-empty basic open W′_{1} ⊂ W_{1} with W′_{1} · x_{1} ⊂ U_{1}. By assumption we find W′_{0} ⊂ W_{0} with (W′_{0}, x_{0}) ≤_{1} (W′_{1}, x_{1}), and hence W′_{0} · x_{0} ⊂ \overline{U_{1}} ⊂ U_{0}. (Claim□)

Definition 6.53. Set x_{0} ≈_{γ} x_{1} if for all i ∈ {0, 1}, β < γ, W_{i} ∈ N, there exists Ŵ_{i} ⊂ W_{i} with Ŵ_{i} ∈ N and W_{1−i} with

(x_{i}, Ŵ_{i}) ≤_{β} (x_{1−i}, W_{1−i}) ≤_{β} (x_{i}, W_{i}).

Note that x_{0}E_{G}^{X}x_{1} implies x_{0} ≈_{γ} x_{1}.

Lemma 6.54. Let B ⊂ X be Π ∼_{α}^{0}, α some countable ordinal. Suppose (x_{0}, V_{0}) ≤_{α} (x_{1}, V_{1}) and x_{1} ∈ B^{∗V_{1}}.

Then x_{0} ∈ B^{∗V_{0}}.

Proof. This is trivial at the base case of α = 1, since it amounts to calculating \overline{V_{0} · x_{0}} and \overline{V_{1} · x_{1}}. Now suppose it is true at all β < α and that

B = ⋂_{i∈ℕ} B_{i}

with each B_{i} ∈ Σ ∼_{β(i)}^{0} some β(i) < α. Then assuming

(x_{0}, V_{0}) ≤_{α} (x_{1}, V_{1})

and i ∈ ℕ and W_{0} ⊂ V_{0} with W_{0} ∈ N. We just need to show that x_{0} ∈ (B_{i})^{ΔW_{0}}. So choose W_{1} ⊂ V_{1} with W_{1} ∈ N and

(x_{0}, W_{0}) ≥_{β(i)} (x_{1}, W_{1}).

Now if x_{0} is not in (B_{i})^{ΔW_{0}}, then for C = X \ B_{i} we have

x_{0} ∈ C^{∗W_{0}}

and so by assumption on β(i) < α we have

x_{1} ∈ C^{∗W_{1}},

with a contradiction to x_{1} ∈ (B_{i})^{ΔW_{1}}.

Corollary 6.55. If every orbit is Π ∼_{α}^{0} then E_{G}^{X} is Π ∼_{α+ω}^{0}.
Proof. x_{0}E_{G}^{X} implies x_{0} ≈_{α+ω} x_{1}. Conversely, x_{0} ≈_{α+ω} x_{1} implies
x_{0} ∈ ([x_{1}]_{G})^{∗G}

by 6.54 and the assumption on the Borel complexity of [x_{1}]_{G}, and hence it implies x_{0}E_{G}^{X}x_{1}. Thus x_{0}E_{G}^{X} if and only if x_{0} ≈_{α+ω} x_{1}, and so the corollary follows 6.51.

In particular, if the orbits are bounded in Borel complexity, then the resulting orbit equivalence relation is Borel, a result obtained previously by Sami in [74] using very different methods.

114 6. A GENERALIZED SCOTT ANALYSIS

CHAPTER 7 GE groups

The GE groups where defined in §6.2 to be the Polish groups such that any Polish G-space with a dense G_{δ} orbit can have only a single dense orbit. They were seen to submit to an especially nice analysis of their turbulent versus classifiable actions.

The central results from this chapter in §7.2 and §7.4 are to the effect that there are many GE groups. They include abelian, even nilpotent, invariantly metrizable, and countable products of locally compact. §7.4 further proves the curious fact that the orbit equivalence relations arising from countable products of locally compact groups all admit classification by countable models. In particular locally compact Polish groups never give rise to turbulent orbit equivalence relations.

§7.3 develops some technology on changing the topologies in a Polish G-space, an idea previewed by the proofs of 6.11 and 6.32. This technique will be used in chapter 9, where a method of transfinite changes in topologies is used to give an analysis of turbulence. The technique is also used in an aside on the Becker dichotomy theorem at §7.5, to the effect that any Polish group admitting a complete left invariant metric group has all its non-smooth orbit equivalence relations ⊑_{c} above E_{0}.

7.1. More on Polish groups; Glimm-Effros; G_{δ} orbits

The results from this section are all well known and mostly classical.

Definition 7.1. Let G be a group. A metric d is said to be left invariant (or right invariant) if for all g, h_{0}, h_{1} ∈ G

d(h_{0}, h_{1}) = d(gh_{0}, gh_{1})

(respectively d(h_{0}, h_{1}) = d(h_{0}g, h_{1}g)). A metric is said to be simply invariant if it is both left and right invariant.

Theorem 7.2. (Birkhoff; Kakutani) Let G be a Polish group. Then G admits a left invariant metric and it admits a right invariant metric.

Proof. It suffices to find a left invariant metric d, since we then obtain a right invariant metric by d_{r}(g, h) = d(g^{−1}, h^{−1}).

Choose (V_{n})_{n∈ℕ} a neighborhood basis of the identity,

V_{n+1}^{4} ⊂ V_{n},
V_{n}^{−1} = V_{n}.

For g_{1}, g_{2} ∈ G let

δ(g_{1}, g_{2}) = inf{2^{−k} : g_{1}^{−1}g_{2} ∈ V_{k}},

and let

d(g_{1}, g_{2}) = inf{∑_{i=0}^{i=k} δ(h_{i}, h_{i+1}) : h_{0} = g_{1}, h_{k+1} = g_{2}}.

Claim: ∀g_{1}, g_{2} ∈ G(d(g_{1}, g_{2}) ≥ δ(g_{1}, g_{2})/2).

Proof of Claim: Suppose instead that there is a counterexample,

δ(g_{1}, g_{2}) = 2^{−n},

with (h_{j})_{j≤k+1}, h_{0} = g_{1}, h_{k+1} = g_{2}, and

∑_{j=0}^{j=k} δ(h_{j}, h_{j+1}) < 2^{−n−1}.

Suppose that k has been chosen as small as possible among counterexamples and over all possible n.

Subclaim: We can find some 0 ≤ i(1, 0) < i(2, 0) ≤ k + 1 so that

δ(h_{i(1,0)}, h_{i(2,0)}) = 2^{−n−1}.

Proof of subclaim: Since by (V_{n+2})^{2} ⊂ V_{n+1} for any j ≤ k + 1 either

δ(h_{0}, h_{j}) ≥ 2^{−n−1}

or

δ(h_{j}, h_{k+1}) ≥ 2^{−n−1}.

So we can either take i(1, 0) = 0, i(2, 0) = j, or else i(1, 0) = j, i(2, 0) = k + 1. (Subclaim□)

Then for any l with i(1, 0) < l < i(2, 0) exactly one of δ(h_{i(1,0)}, h_{l}), δ(h_{l}, h_{i(2,0)}) is equal to 2^{−n−1} – at least one else we obtain δ(h_{i(1,0)}, h_{i(2,0)}) ≤ 2^{−n−1} by (V_{n+2})^{4} ⊂ V_{n+1}, and at most one since for all

m_{1} < m_{2} ≤ m_{3} < m_{4} ≤ k + 1

the minimality of k implies

d(h_{m_{1}}, h_{m_{2}}) ≥ δ(h_{m_{1}}, h_{m_{2}})/2

and

d(h_{m_{3}}, h_{m_{4}}) ≥ δ(h_{m_{3}}, h_{m_{4}})/2,

and hence by assumption of

∑_{j=0}^{j=k} d(h_{j}, h_{j+1}) < 2^{−n−1}

either

δ(h_{m_{1}}, h_{m_{2}}) < 2^{−n−1}

or

δ(h_{m_{3}}, h_{m_{4}}) < 2^{−n−1}.

Repeating this we may successively find i(1, k) and i(2, k) at each k so that {i(1, k), i(1, k)+1, ..., i(2, k)} strictly includes {i(1, k+1), i(1, k+1)+1, ..., i(2, k+1)} and

δ(h_{0}, h_{i(1,k)}) < 2^{−n−1},
δ(h_{i(1,k)}, h_{i(2,k)}) = 2^{−n−1},
δ(h_{i(2,k)}, h_{k+1}) < 2^{−n−1}.

7.1. MORE ON POLISH GROUPS; GLIMM-EFFROS; G_{δ} ORBITS 117

At last we arrive at some n with i(2, k) = i(1, k) + 1, with a contradiction. (Claim □)

By V_{n} = V_{n}^{−1} for all g_{1}, g_{2} ∈ G
d(g_{1}, g_{2}) = d(g_{2}, g_{1}).

The structure of the definition of d enforces the triangle inequality. By the claim d(g_{1}, g_{2}) = 0 implies g_{1} = g_{2}. So d is a metric.

From the claim and the definition d is compatible with the topology on G.

Theorem 7.3. (folklore) Let G be a Polish group with {U_{n} : n ∈ ℕ} a neighborhood basis of the identity consisting of conjugation invariant sets – so that for all g ∈ G and n ∈ ℕ

gU_{n}g^{−1} = U_{n}.

Then G admits an invariant metric.

Proof. First by replacing U_{n} by U_{n}∩U_{n}^{−1} we may assume that each U_{n} = U_{n}^{−1}. Then we may choose {V_{n} : n ∈ ℕ} ⊂ {U_{n} : n ∈ ℕ} with V_{n+1} chosen sufficiently small that (V_{n+1})^{4} ⊂ V_{n}.

Now proceeding through the construction of 7.2 we obtain for all g_{1}, g_{2}, h ∈ G that

δ(g_{1}, g_{2}) = inf{2^{−k} : g_{1}^{−1}g_{2} ∈ V_{k}} = inf{2^{−k} : g_{1}^{−1}g_{2} ∈ hV_{k}h^{−1}},

since hV_{k}h^{−1} = V_{k}, and hence in turn equals inf{2^{−k} : h^{−1}g_{1}^{−1}g_{2}h ∈ V_{k}} = δ(g_{1}h, g_{2}h), so the metric is not just left invariant but also right invariant.

The conclusion of 7.3 implies the assumption, since if d is an invariant metric, then for each n we may let U_{n} = {g ∈ G : d(g, 1_{G}) < 2^{−n}}.

Lemma 7.4. Any compatible invariant metric on a Polish group is complete.

Proof. Let G be the Polish group, d the invariant metric, X the Cauchy completion of (G, d). G is a G_{δ} subset of X, since if d̂ is a complete metric on G, {g_{i} : i ∈ ℕ} dense in G, and for each n, i we choose open O_{i,n} ⊂ X with

O_{i,n} ∩ G = {g ∈ G : d̂(g, g_{i}) < 2^{−n}}

then G = {x ∈ X : ∀n∃i(x ∈ O_{i,n}}.

X is a Polish space, since the separability of G imposes separability on X. In fact it may be naturally viewed as a Polish G-space.

For g_{i} → x a Cauchy sequence, x ∈ X, then we can define g·x to be the Cauchy limit of (g · g_{i})_{i∈ℕ}; this is well defined by left invariance of d. We obtain that the action is continuous – for if g_{i} → x and h_{i} → y as i → ∞, then for all g, h ∈ G

d(g · x, h · y) = lim_{i→∞}d(g · g_{i}, h · h_{i}) ≤ lim_{i→∞}d(g_{i}, g^{−1}h · h_{i})
≤ lim_{i→∞}d(g_{i}, h_{i}) + d(h_{i}, g^{−1}h · h_{i})

which by right invariance of the metric equals

lim_{i→∞}d(g_{i}, h_{i}) + d(h, g).

If we now manage to show that X has only one orbit we will license the conclusion X = G.

So fix x ∈ X the Cauchy limit of (h_{i})_{i∈ℕ}. Note that [x]_{G} is dense, since for g ∈ G with d(x, g) < ϵ one has for all h ∈ G that d(hg^{−1}x, h) < ϵ by left invariance of d. Note that every x̂ ∈ [x]_{G} will be the Cauchy limit of (ĝh_{i})_{i∈ℕ} for some ĝ ∈ G.

Note that if x̂_{0}, x̂_{1} are the Cauchy limits of (ĝ_{0}h_{i})_{i∈ℕ}, (ĝ_{1}h_{i})_{i∈ℕ} respectively, then d(x̂_{0}, x̂_{1}) = d(g_{0}, g_{1}). Define

π : [x]_{G} → G
g · x ↦ g.

By right invariance π is a well defined and a homeomorphism – and indeed not just a homeomorphism but an isometry.

Thus for any x_{1}, x_{2} ∈ X the orbits [x_{1}]_{G} and [x_{2}]_{G} are dense G_{δ}’s and so have a point in common.

Theorem 7.5. (Hausdorff) Let X be Polish and Y metrizable. If π : X ↠ Y is open, continuous, and onto, then Y is Polish.

Proof. Let Ŷ be the Cauchy completion of Y. Since Y separable, Ŷ is Polish, and so by 2.3(i) it suffices to show Y is a G_{δ} in Ŷ.

Define open N_{s} ⊂ X for s ∈ ℕ^{<ℕ} by induction on the length of s. N_{∅} = X. {N_{s} : s ∈ ℕ^{1}} will cover X with each N_{s} having diameter less than 1. Given {N_{s} : s ∈ ℕ^{≤k}} we then choose {N_{s} : s ∈ ℕ^{k+1}} so that at each t ∈ ℕ^{k}

N_{t} = ⋃{N_{s} ∈ ℕ^{k+1} : s ⊃ t}

and the diameter of N_{t} is less than 2^{−k−1} with respect to the complete metric in X. We also insist that

\overline{N_{t}} ⊂ N_{s}

for s strictly included in t.

Then choose open W_{s} ⊂ Ŷ open for s ∈ ℕ^{<ℕ} so that π[N_{s}] ∩ Y = W_{s} ∩ Y. Using that open sets are F_{σ} we can obtain open V_{s} so that:

(i) ⋃{V_{s} : s ∈ ℕ^{1}} = ⋃{W_{s} : s ∈ ℕ^{1}};

(ii) for all s ∈ ℕ^{<ℕ}

V_{s} ⊂ W_{s};

(iii) for all k ∈ ℕ, s ∈ ℕ^{k}

⋃{V_{t} : t ∈ ℕ^{k+1}, t ⊃ s} = V_{s} ∩ ⋃{W_{t} : t ∈ ℕ^{k+1}, t ⊃ s};

(iv) for all y ∈ Ŷ, k ∈ ℕ, and s ∈ ℕ^{k}, there are only finitely many t ⊃ s in ℕ^{k+1} with y ∈ V_{t}, and there are only finitely many t ∈ ℕ^{1} with y ∈ V_{t}.

(iv) is the only difficulty in all this, but if each W_{t} = ⋃ C_{n,t} with each C_{n,t} closed, C_{n,t} ⊂ C_{n+1,t}, then we can demand for t_{0}, t_{1} ∈ ℕ^{k+1}, s ∈ ℕ^{k}, s ⊂ t_{0}, t_{1}, if

t_{0}(k) < t_{1}(k)

then

W_{t_{1}} ∩ C_{n,t_{0}} = ∅

for n < t_{1}(k). Consequently for each y ∈ Ŷ and k ∈ ℕ there are only finitely many s ∈ ℕ^{k} with y ∈ V_{s}.

All this done, it is clear that

Y ⊂ ⋂_{k∈ℕ} ⋃_{s∈ℕ^{k}} V_{s}.

So it is enough that:

Claim: Y ⊃ ⋂_{k∈ℕ} ⋃_{s∈ℕ^{k}} V_{s}.

Proof of Claim (this is König’s infinity lemma). Fix y ∈ ⋂_{k∈ℕ} ⋃_{s∈ℕ^{k}} V_{s} and let

T_{y} = {s ∈ ℕ^{<ℕ} : there exists infinitely many t ⊃ s with y ∈ V_{t}}.

T_{y} is a tree, in the sense that

s ⊂ t ∈ T_{y}

implies

s ∈ T_{y}.

y ∈ V_{s} whenever s ∈ T_{y} by (iii). Since there can be only finitely many s ∈ ℕ^{1} for which y ∈ V_{s} the set T_{y} is non-empty, and by (iv) for each s_{0} ∈ T_{y} there must exist s_{1} ∈ T_{y} such that s_{1} ⊃ s_{0} and s_{1} ≠ s_{0} Thus we find f ∈ ℕ^{ℕ} such that for each k ∈ ℕ

f|_{k} ∈ T_{y}
∴ y ∈ V_{f|_{k}}.

We may also find

x ∈ ⋂_{k∈ℕ} N_{f|_{k}}

by the completeness of the metric. Since {N_{f|_{k}} : k ∈ ℕ} is a neighborhood basis of x and the map is continuous it follows that y is in every open set containing π(x) and thus y = π(x). (Claim□)

Corollary 7.6. If G be a Polish group and H is a closed subgroup then G/H in the quotient topology is Polish.

Proof. If d is a right invariant metric on G then

d(g_{0}H, g_{1}H) = inf{d(h_{0}, h_{1}) : h_{0} ∈ g_{0}H, h_{1} ∈ g_{1}H}

provides a compatible metric on G/H, and so the quotient is Polish.

In 7.6 we have that G/H is a Polish G-space under left translation.

The importance of GE groups is indicated by the next few results, the first of which is presented in [4] as an abstraction of [25] and [14]. Recall E_{0} as the equivalence relation of eventual agreement on 2^{ℕ}, the Cantor space of infinite binary sequences.

Theorem 7.7. (Becker, Kechris) Let G be a Polish group and X a properly generically ergodic Polish G-space – that is to say, every orbit dense and meager. Then

E_{0} ⊑_{c} E_{G}^{X}.

Proof. By 2.46 and the fact that E_{G}^{X} as a Σ ∼_{1}^{1} subset of the product has the Baire property, the orbit equivalence relation must be meager in X × X; so find {O_{i} : i ∈ ℕ} open dense sets in X × X witnessing this:

E_{G}^{X} ∩ ⋂_{i∈ℕ} O_{i} = ∅.

Assume that O_{n} ⊂ O_{m} whenever n ≥ m. For s ∈ 2^{<ℕ} build non-empty open sets V_{s} ⊂ X and group elements {g_{s,t} : s, t ∈ 2^{k}} ⊂ G for each k ∈ ℕ, so that:

(i) d(V_{s}) < 2^{−k}, for s ∈ 2^{k}, where d is the complete metric on X; s ⊃ t and s ≠ t implies

\overline{V_{t}} ⊂ V_{s};

(ii) for s, t ∈ 2^{k} some k,
g_{s,t} · V_{s} = V_{t};
(iii) for s, t ∈ 2^{k} with s(n) ≠ t(n),
V_{s} × V_{t} ⊂ O_{n},
V_{s} ∩ V_{t} = ∅;

(iv) if for s ∈ 2^{k} and j ∈ {0, 1} we define s^{⌢}j ∈ 2^{k+1} by the requirement

that s ⊂ s^{⌢}j, s^{⌢}j(k) = j, then for all k ∈ ℕ and s, t ∈ 2^{k}
g_{s^{⌢}0,t^{⌢}0} = g_{s^{⌢}1,t^{⌢}1} = g_{s,t}.

Granting that this may be done, and running on to the finish, note that we could then define θ : 2^{ℕ} → X by letting θ(y) be the unique point in

⋂{V_{s} : s ∈ 2^{<ℕ}, y ⊃ s}.

This is well defined and continuous by (i); one to one by (iii). If y_{0}, y_{1} ∈ 2^{ℕ} with y_{0}E_{0}y_{1}, then we come to some N ∈ ℕ such that for all k ≥ N in ℕ

y_{0}(k) = y_{1}(k)

and hence for all k ≥ N

g_{y_{0}|_{k},y_{1}|_{k}} = g_{y_{0}|_{N} ,y_{1}|_{N}}

by (iv), and hence

g_{y_{0}|_{N} ,y_{1}|_{N}} · θ(y_{0}) = θ(y_{1})

by (ii) and continuity of the action. Alternatively if there are infinitely many k such that y_{0}(k) ≠ y_{1}(k) then are infinitely many k with

(θ(y_{0}), θ(y_{1})) ∈ O_{k}

by (iii), and hence

(θ(y_{0}), θ(y_{1})) ∈ ⋂_{k∈ℕ} O_{k}

by the choice of (O_{k})_{k∈ℕ} as a decreasing sequence, and so θ(y_{0}) and θ(y_{1}) are orbit inequivalent.

To perform this construction begin with W_{⟨0⟩}^{0}, W_{⟨1⟩}^{0} open nonempty subsets of X with diameter less than 1/2 and

W_{⟨0⟩}^{0} × W_{⟨1⟩}^{0} ⊂ O_{0}.

Refining find W_{⟨0⟩}^{1} ⊂ W_{⟨0⟩}^{0} and W_{⟨1⟩}^{1} ⊂ W_{⟨1⟩}^{0} with

W_{⟨0⟩}^{1} × W_{⟨1⟩}^{1} ⊂ O_{0}.

By every orbit dense we can find x ∈ W_{⟨0⟩}^{1}, g ∈ G with g · x ∈ W_{⟨1⟩}^{1}. By continuity of the action let V_{⟨0⟩} ⊂ W_{⟨0⟩}^{1} be open containing x with g · V_{⟨0⟩} ⊂ W_{⟨1⟩}^{1}. Let g_{⟨0⟩,⟨1⟩} = g and V_{⟨1⟩} = g · V_{⟨0⟩}. (g_{⟨0⟩,⟨0⟩} and g_{⟨1⟩,⟨1⟩} are taken to be the identity; g_{⟨1⟩,⟨0⟩} = g_{⟨0⟩,⟨1⟩}^{−1}.)

At the next step note that we may first choose disjoint W_{⟨0,1⟩}^{0}, W_{⟨0,0⟩}^{0} ⊂ V_{⟨0⟩} that are non-empty and open having diameter less than 2^{−2}

W_{⟨0,0⟩}^{0} × W_{⟨0,1⟩}^{0} ⊂ O_{1};

V_{⟨1⟩}  g_{⟨0⟩,⟨1⟩}  V_{⟨0⟩}

Figure 7.1. Production of V_{⟨0⟩} and V_{⟨1⟩}

this uses O_{1} open dense. Using the same consideration find non-empty W_{⟨0,0⟩}^{1} ⊂ W_{⟨0,0⟩}^{0} W_{⟨0,1⟩}^{1} ⊂ W_{⟨0,1⟩}^{0} with

g_{⟨0⟩,⟨1⟩} · W_{⟨0,0⟩}^{1} × g_{⟨0⟩,⟨1⟩} · W_{⟨01⟩}^{1} ⊂ O_{1};

this can be done, since O_{1} is open dense. Going again find W_{⟨0,0⟩}^{2} ⊂ W_{⟨0,0⟩}^{1} and W_{⟨0,1⟩}^{2} ⊂ W_{⟨0,1⟩}^{1} such that

W_{⟨0,1⟩}^{2} × W_{⟨0,0⟩}^{2} ⊂ O_{1};

then W_{⟨0,0⟩}^{3} ⊂ W_{⟨0,0⟩}^{2} and W_{⟨0,1⟩}^{3} ⊂ W_{⟨0,1⟩}^{2} such that

g_{⟨0⟩,⟨1⟩} · W_{⟨0,1⟩}^{3} × g_{⟨0⟩,⟨1⟩} · W_{⟨00⟩}^{3} ⊂ O_{1};

and again for W_{⟨0,0⟩}^{4} ⊂ W_{⟨0,0⟩}^{3} and W_{⟨0,1⟩}^{4} ⊂ W_{⟨0,1⟩}^{3} so

W_{⟨0,0⟩}^{4} × g_{⟨0⟩,⟨1⟩} · W_{⟨0,1⟩}^{4} ⊂ O_{1};

and again W_{⟨0,0⟩}^{5} ⊂ W_{⟨0,0⟩}^{4} and W_{⟨0,1⟩}^{5} ⊂ W_{⟨0,1⟩}^{4} so

g_{⟨0⟩,⟨1⟩} · W_{⟨0,1⟩}^{5} × W_{⟨0,0⟩}^{5} ⊂ O_{1};

again for W_{⟨0,0⟩}^{6} ⊂ W_{⟨0,0⟩}^{5} and W_{⟨0,1⟩}^{6} ⊂ W_{⟨0,1⟩}^{5} so

W_{⟨0,1⟩}^{6} × g_{⟨0⟩,⟨1⟩} · W_{⟨0,0⟩}^{6} ⊂ O_{1};

and again W_{⟨0,0⟩}^{7} ⊂ W_{⟨0,0⟩}^{6} and W_{⟨0,1⟩}^{7} ⊂ W_{⟨0,1⟩}^{6} so

g_{⟨0⟩,⟨1⟩} · W_{⟨0,0⟩}^{7} × W_{⟨0,1⟩}^{7} ⊂ O_{1};

Next let

W_{⟨0,0⟩} = W_{⟨0,0⟩}^{7},
W_{⟨0,1⟩} = W_{⟨0,1⟩}^{7},
W_{⟨1,0⟩} = g_{⟨0⟩,⟨1⟩} · W_{⟨0,0⟩}^{7},
W_{⟨1,1⟩} = g_{⟨0⟩,⟨1⟩} · W_{⟨0,1⟩}^{7}.

We may assume that the diameters of W_{⟨1,1⟩} and W_{⟨1,0⟩} are also less than 2^{−2}, since at worst this only requires us to shrink W_{⟨0,0⟩}^{7} and W_{⟨0,1⟩}^{7} one more time.

Using the dense orbits we find V_{⟨0,0⟩} ⊂ W_{⟨0,0⟩} and g_{⟨0,0⟩,⟨0,1⟩} ∈ G with

g_{⟨0,0⟩,⟨0,1⟩} · V_{⟨0,0⟩} ⊂ W_{⟨0,1⟩}.

Let

V_{⟨0,1⟩} = g_{⟨0,0⟩,⟨0,1⟩} · V_{⟨0,0⟩}.

V_{⟨1⟩}  V_{⟨0⟩} V_{⟨10⟩}  g_{⟨0⟩,⟨1⟩}  V_{⟨00⟩} V_{⟨11⟩}  V_{⟨01⟩} g_{⟨10⟩,⟨11⟩}  g_{⟨00⟩,⟨01⟩}
Figure 7.2. Subdividing V_{⟨0⟩} and V_{⟨1⟩}

After this the definitions

V_{⟨1,0⟩} = g_{⟨0⟩,⟨1⟩} · V_{⟨0,0⟩}

and

V_{⟨1,1⟩} = g_{⟨0⟩,⟨1⟩} · V_{⟨0,1⟩}

are both necessary and good. The group elements at the second level are defined by the requirement that g_{⟨0⟩,⟨1⟩} copies the arrangement of (V_{⟨0,0⟩}, V_{⟨0,1⟩}) over to (V_{⟨1,0⟩}, V_{⟨1,1⟩}) and that in general g_{s,t} = (g_{t,s})^{−1}:

g_{⟨0,1⟩,⟨0,0⟩} = g_{⟨0,0⟩,⟨0,1⟩}^{−1};
g_{⟨0,1⟩,⟨1,1⟩} = g_{⟨0,0⟩,⟨1,0⟩} = g_{⟨0⟩,⟨1⟩}

and

g_{⟨1,1⟩,⟨0,1⟩} = g_{⟨1,0⟩,⟨0,0⟩} = g_{⟨1⟩,⟨0⟩} = g_{⟨0⟩,⟨1⟩}^{−1};
g_{⟨0,0⟩,⟨1,1⟩} = g_{⟨0⟩,⟨1⟩}g_{⟨0,0⟩,⟨0,1⟩};
g_{⟨1,0⟩,⟨1,1⟩} = g_{⟨0⟩,⟨1⟩}g_{⟨0,0⟩,⟨0,1⟩}g_{⟨0⟩,⟨1⟩}^{−1}.

This completes the construction for s ∈ 2^{<ℕ} of length of at most two. Continuing in this fashion we finish.

In virtue of 3.4 theorem 7.7 may be equivalently reformulated as

Corollary 7.8. Let G be a Polish group and X a Polish G-space. (i) If every orbit meager and at least one orbit dense then

E_{0} ⊑_{c} E_{G}^{X}.

(ii) If there is some orbit [x]_{G} so that for no G_{δ} orbit [y]_{G} do we have

\overline{[x]_{G}} = \overline{[y]_{G}}

then

E_{0} ⊑_{c} E_{G}^{X}.

Proof. (i) follows directly from 7.7 and 3.5 and that any non-meager subset of the space X_{0} described there must be non-meager in X.

For (ii): Suppose we may find x ∈ X for which there is no y with [x]_{G} a G_{δ} and \overline{[x]_{G}} = \overline{[y]_{G}}. Let

X(x) = \{z ∈ X : \overline{[x]_{G}} = \overline{[z]_{G}}\}.

X(x) is a G_{δ} since if {U_{l} : l ∈ ℕ} forms a basis of X and {g_{i} : i ∈ ℕ} is dense in G then

\overline{[x]_{G}} = \overline{[z]_{G}}

if and only if

∀i, l, ∃j(g_{i} · x ∈ U_{l} ⇒ g_{j} · z ∈ U_{l})

and

∀i, l, ∃j(g_{i} · z ∈ U_{l} ⇒ g_{j} · x ∈ U_{l}).

Since the class of Polish spaces is closed under taking G_{δ} subsets by 2.3, we have that X(x) is a Polish G-space. Since no orbit in X(x) can be G_{δ}, every orbit is meager by 6.37. Thus by 7.7

E_{0} ⊑_{c} E_{G}^{X(x)} ⊑_{c} E_{G}^{X}.

Theorem 7.9. Let G be a GE group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E_{G}^{X}.

Proof. Claim: At most one of 1. and 2.

Proof of claim: It suffices to observe that E_{0} is not Borel reducible to id(ℝ). To prove this note that E_{0} is induced by the natural action of (ℤ_{2})^{<ℕ} (the subgroup of (ℤ_{2})^{ℕ} consisting of g⃗ that has g⃗(i) = 0 for all sufficiently large i) on 2^{ℕ} under identification of ℤ_{2} and {0, 1} – namely, for g⃗ ∈ (ℤ_{2})^{<ℕ} and x⃗ ∈ 2^{ℕ} we let (g⃗·x⃗)(i) = g⃗(i) + x⃗(i) mod 2. Since every orbit under this action is dense and countable, and so certainly meager, we can appeal to 3.3. (Claim □)

To show that at least one holds, first suppose that for all x, y ∈ X the closure of the two orbits are equal if and only if the orbits themselves are equal. Fix {g_{i} : i ∈ ℕ} a dense subset of G and {U_{l} : l ∈ ℕ} a basis. To each point x ∈ X assign the real θ(x) in the unit interval whose lth coordinate in its decimal expansion is 5 if there exists some i for which g_{i} · x ∈ U_{l} and 6 otherwise: so if A_{l} = G · U_{l} then

θ : X → [0, 1]
θ(x) = 0 · χ_{A_{0}}(x)χ_{A_{1}}(x)χ_{A_{2}}(x)...

witnesses the truth of (I).

So now suppose there are x, y ∈ X with

\overline{[x]_{G}} = \overline{[y]_{G}}

but

[x]_{G} ≠ [y]_{G}.

Then by the definition of a GE group there can be no G_{δ} orbit with the same closure as [x]_{G} and we have E_{0} ⊑_{c} E_{G}^{X} by 7.8.

This gives a non historical proof of

Theorem 7.10. (Effros) Let G be a locally compact group and X a Polish Gspace. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E_{G}^{X}.

Proof. It suffices to show that locally compact entails GE. But if X_{0} is a Polish G-space with every orbit dense and [x]_{G} is G_{δ} it must also be that

[x]_{G} ∈ F_{σ},

since G is locally compact and g ↦ g · x is continuous; then given any y ∈ X which is E_{G}-inequivalent to x we obtain that both [x]_{G} and its complement are dense G_{δ}’s in X_{0} with a contradiction to the Baire category theorem.

Lemma 7.11. (Effros) Suppose that G is a Polish group and X is a Polish G-space. Then the following are equivalent:

(I) every orbit is G_{δ}; (II) for all x, y ∈ X,

[x]_{G} = [y]_{G} ⇔ \overline{[x]_{G}} = \overline{[y]_{G}};

(III) E_{G}^{X} is G_{δ}. Moreover each of these implies that E_{G}^{X} is smooth.

Proof. (I)⇒(II): Since disjoint G_{δ} sets cannot have the same closure by the Baire category theorem.

(II)⇒(III): Since for {U_{l} : l ∈ ℕ} a basis of X and {g_{i} : i ∈ ℕ} a dense subgroup of G we have \overline{[x]_{G}} = \overline{[y]_{G}} if and only if

∀i, l, ∃j(g_{i} · x ∈ U_{l} ⇒ g_{j} · y ∈ U_{l})

and

∀i, l, ∃j(g_{i} · y ∈ U_{l} ⇒ g_{j} · x ∈ U_{l}),

both of which are G_{δ}.

(III)⇒(I). Fix x ∈ X and suppose

E_{G}^{X} = ⋂_{n∈ℕ} O_{n},

each O_{n} open in X × X. Then for O_{n}^{x} = {y : (x, y) ∈ O_{n}}, each O_{n}^{x} is open and

[x]_{G} = ⋂_{n∈ℕ} O_{n}^{x}.

Note that (II) implies E_{G}^{X} ≤_{B} id(ℝ), since by the argument of 7.9 we can then assign to each point a code for the closure of its orbit.

Theorem 7.12. (Effros) Suppose that G is a Polish group, X is a Polish Gspace, and x ∈ X. Then the following are equivalent:

(I) the map

G → [x]_{G}
g ↦ g · x

is open;

(II) [x]_{G} ∈ G_{δ};

(III) [x]_{G} is non-meager in \overline{[x]_{G}}.

Proof. (I)⇒(II): For then if G_{x} =_{df} {g ∈ G : g · x = x} we have that G/G_{x} and [x]_{G} are homeomorphic, and then 7.6 implies both are Polish spaces, and thus [x]_{G} ∈ G_{δ}.

(II)⇒(I): (II) implies that [x]_{G} is itself a Polish G-space. Define

f : [x]_{G} → G/G_{x}

by letting f(x′) be the unique g′G_{x} in G/G_{x} with g′ · x = x′. The space G/G_{x} is Polish by 7.6. It is enough to show that f is continuous.

For any V ⊂ G open the set {x′ ∈ [x]_{G} : ∃g′ ∈ V (g′ · x = x′)} is Σ ∼_{1}^{1}, and so f is Baire measurable by 2.52. Choose C ⊂ [x]_{G} comeager on which f is continuous by 2.49. By 2.47 and the fact that G acts transitively on the Polish space [x]_{G} we have that for all x′ ∈ [x]_{G}

∀^{∗}g ∈ G(g · x′ ∈ C).
Thus for any x_{0}, x_{1}, x_{2}, ..., x_{∞} ∈ [x]_{G} with
x_{i} → x_{∞}

we may find g ∈ G so that g · x_{∞} ∈ C and each g · x_{i} ∈ C. Thus if g_{∞} · x = x_{∞} and at each i we choose g_{i} with g_{i} · x = x_{i} then it follows from the definition of C that

gg_{i}G_{x} → gg_{∞}G_{x}

and therefore

g_{i}G_{x} → g_{∞}G_{x},

which amounts to asserting that

f(x_{i}) → f(x_{∞})

just as required.

(III)⇒(II): This is 6.37.

(II)⇒(III): Dense G_{δ}’s are by definition comeager.

As a word towards notation, the term GE group derives from the following definition of [42]:

Definition 7.13. A Polish group G has the strong Glimm-Effros property if whenever X is a Polish G-space exactly one of the following hold:

1. E_{0} ⊑_{c} E_{G}^{X};

2. E_{G}^{X} is G_{δ}.

By now it is not much of a surprise that:

Lemma 7.14. Any GE group has the strong Glimm-Effros property.

Proof. Let X be a Polish G-space. If there are two distinct orbits with the same closure then we obtain E_{0} ⊑_{c} E_{G}^{X} as in the proof of 7.9. Otherwise we obtain that the closure of an orbit classifies the orbit completely and appeal to 7.11.

It is unknown whether GE groups are the only groups with the strong Glimm-Effros property.

Exercise 7.15. Any right invariant metric on a locally compact Polish group is necessarily complete.

Exercise 7.16. If one compatible right invariant metric on a Polish group is complete then they all are.

Exercise 7.17. If a compact Polish group acts continuously on a Polish space then the orbit equivalence relation is smooth.

Solecki has recently proved a converse: If a Polish group G has E_{G}^{X} smooth for every Polish G-space X then G is compact. Kechris shows in [49] that every locally compact Polish group has all its orbit equivalence relations ≤_{B} some Borel equivalence relation with countable equivalence classes; he goes on to conjecture its converse:

If a Polish group G is such that whenever it acts continuously on a Polish space X we have E_{G}^{X} ≤_{B} E some Borel equivalence relation with all classes countable, then G is locally compact.

This remains stubbornly open.

Exercise 7.18. (See [4], 2.3.5(b).) Let H be a closed subgroup of a Polish group Ĥ. Let X be a Polish H-space. With the following we may lift the action of H on X to an action Ĥ on X̂ so that X is a closed subspace of X̂ and each orbit [x]Ĥ meets exactly one orbit in X.

Let X̂ be the set of all [x, g] with x ∈ X and g ∈ Ĥ where we set [x, g] = [x′, g′] if there is some h ∈ H with h · x = x′ and hg = g′; give X̂ the quotient topology provided by the surjection

X × Ĥ ↠ X̂
(x, g) ↦ [x, g].

(i) Show that X̂ is Hausdorff.

(ii) Show that for any open U ⊂ X̂ and z ∈ U we may find open U_{0} containing z with \overline{U_{0}} ⊂ U.

(iii) Use the separability of X̂ and Urysohn’s metrizability criterion at §8.3 [73] to conclude that X̂ is metrizable; hence by 7.5 it is Polish.

(iv) In the action

g · [x, g′] = [x, g′g^{−1}]

X̂ is a Polish Ĥ-space.

(v) X may be identified with {[x, 1] : x ∈ X}, so that E_{H}^{X} equals the restriction

of EX̂ ; Ĥ to X.

[53] shows this lift to preserve turbulence, so if X is a turbulent Polish H-space then X̂ is a turbulent Ĥ-space.

Exercise 7.19. (Gao [22]) If H is Polish and G is a closed normal subgroup with both G and H/G admitting a complete left invariant metric, then H admits a complete left invariant metric, and thus all solvable Polish groups have complete left invariant metrics.

Exercise 7.20. A homogeneous compact metric space is one on which the homeomorphism group acts transitively. Use 7.12 to show that if (K, d) is a homogeneous compact metric space then for all ϵ > 0 there is δ > 0 such that for all x, y ∈ X with d(x, y) < δ there is some π ∈ Hom(K) such that

π(x) = y

and for all z ∈ K

d(π(z), z) < ϵ.

This abrupt consequence of Effros’ theorem turned out to be profoundly useful in [28].

Exercise 7.21. (Compare §6.3 of [4] and the proofs of 6.11 and 6.17 above.)

(i) Suppose M ∈ Mod(L). Show that the following three things are equivalent: (a) [M]_{S_{∞}} ∈ G_{δ} (with respect to the usual topology on Mod(L)).

(b) For all n_{0}, n_{1}, ..., n_{k} ∈ ℕ^{<ℕ} there is a quantifier free formula φ such that

M ⊧ ∃z⃗φ(z⃗, n_{0}, n_{1}, ..., n_{k})

and for all formulas ψ(x_{0}, ..., x_{k}) in the language we have either

M ⊧ ∀x_{0}...∀x_{k}(∃z⃗φ(z⃗, x⃗) ⇒ ψ(x⃗))

or

M ⊧ ∀x_{0}...∀x_{k}(∃z⃗φ(z⃗, x⃗) ⇒ ¬ψ(x⃗)).

(c) For all n_{0}, n_{1}, ..., n_{k} ∈ ℕ^{<ℕ} there is a quantifier free formula φ such that

M ⊧ ∃z⃗φ(z⃗, n_{0}, n_{1}, ..., n_{k})

and for all quantifier free formulas ψ(x_{0}, ..., x_{k}, y_{0}, ..., y_{l}) in the language we have either

M ⊧ ∀x_{0}...∀x_{k}(∃z⃗φ(z⃗, x⃗) ⇒ ∃y_{0}...∃y_{l}ψ(x⃗, y⃗))

or

M ⊧ ∀x_{0}...∀x_{k}(∃z⃗φ(z⃗, x⃗) ⇒ ∀y_{0}...∀y_{l}¬ψ(x⃗, y⃗)).

(ii) (Miller; Suzuki) Let τ_{F O} be the Polish topology from 2.43. Show that the following are equivalent:

(a) [M]_{S_{∞}} is G_{δ} with respect to the τ_{F O}.

(b) M is an atomic model; in other words, for all n_{0}, n_{1}, ..., n_{k} ∈ ℕ^{<ℕ} there is some formula φ such that

M ⊧ φ(n_{0}, n_{1}, ..., n_{k})

and for all ψ(x_{0}, ..., x_{k}) in the language we have either

M ⊧ ∀x_{0}...∀x_{k}(φ(x⃗) ⇒ ψ(x⃗))

or

M ⊧ ∀x_{0}...∀x_{k}(φ(x⃗) ⇒ ¬ψ(x⃗)).

(Hint for (i): For (c)⇒(a), show that γ^{∗}(M) = 0 and appeal to 6.20 and 7.12. For (a)⇒(b), we can let V = {g ∈ S_{∞} : g(n_{i}) = n_{i} all i ≤ k}; quoting the definition of the topology and 7.12 we have some quantifier free φ as above)

Clearly there is some kind of combinatorial difference in (i)(b) and (ii)(b), since for the topology generated by quantifier free formulas one does not have the set {N : ∃z⃗φ(z⃗, n_{0}, ..., n_{k})} as a clopen set even for φ quantifier free. An important step in the proof of Becker’s theorem at §7.5 is to pass to a topology which approximates this property of τ_{F O}; we do not quite obtain for any basic open U ⊂ X and V ⊂ G that U^{ΔV} is clopen, but rather the approximation indicated at 7.32 and 7.48(III).

Exercise 7.22. (The Vitali equivalence relation produced by rational translation is equireducible with E_{0}. See also [13].)

(i) E_{0} ⊑_{c} ℝ/ℚ – that is to say, there is a continuous injection reducing E_{0} to the orbit equivalence relation resulting from ℚ acting by translation on ℝ.

(ii) Let G be the group ℚ/ℤ obtained by taking the algebraic quotient of ℚ by the subgroup ℤ; let

π_{1} : ℚ ↠ G

be the quotient map. Let 𝕋 be the space ℝ/ℤ obtained by taking the topological quotient of ℝ by the closed equivalence relation {(x, y) ∈ ℝ^{2} : x − y ∈ ℤ}; let

π_{2} : ℝ ↠ 𝕋

be the canonical surjection.

We can then let G act of 𝕋 in the obvious manner:

(π_{1}(q)) · (π_{2}(x)) = π_{2}(q + x).

One then has

ℝ/ℚ ≤_{B} 𝕋/G.

(iii) G may be written as

G = ⋃_{n∈ℕ} G_{n}

where each G_{n} is a finite subgroup of G and G_{n} ⊂ G_{n+1}.

(iv) Choose {I_{s} : s ∈ ℕ^{<ℕ}} half-open intervals in 𝕋 such that I_{∅} = 𝕋, each I_{s} is the disjoint union of (I_{s^{⌢}i})_{i∈ℕ}, d(I_{s}) approaches 0 as the length of s approaches ∞, for some choice of a compatible metric d, and for each s ∈ ℕ^{k} and i ∈ ℕ

G_{k} · I_{s^{⌢}i} ∩ I_{s} = I_{s^{⌢}i}.
(v) Let A be the set ℕ^{<ℕ}. For each x ∈ 𝕋 define θ(x) ∈ 2^{A} by
(θ(x))(s) = 1

if and only if the length of s equals n + 1 some n and G_{n} · x ∩ I_{s} ≠ ∅ where n + 1. For x, y ∈ 𝕋 we have that θ(x) and θ(y) agree off of a finite set if and only if for some n

G_{n} · x = G_{n} · y.

(vi) Thus by (v) and (ii)

ℝ/ℚ ≤_{B} E_{0}.

7.2. Invariantly metrizable and nilpotent are GE

Both these results appear in [42]. The first to be proved, and even then initially in the special case of G abelian, was 7.23 and at the time it was very surprising.

The point is that the we should not be so shocked that smoothness for locally compact groups implies every orbit is G_{δ}. Local compactness of the group is sufficient to imply every orbit F_{σ}, and so the further reduction to the class G_{δ} ∩ F_{σ} is a small one.

However the situation with abelian Polish groups was already known from Solecki’s work at [76] to be completely different. Abelian Polish groups are capable of giving rise to orbit equivalence relations that are not even Borel. Thus part of the content of the next theorem is to say that when an orbit equivalence relation induced by an abelian Polish group action is smooth we necessarily have an enormous reduction in the potential Borel complexity. This was the first result of its kind.

Theorem 7.23. (Solecki) Any Polish group admitting an invariant metric is GE.

Proof. Let d be an invariant metric on a Polish group G, which in turn will be complete by 7.4, and let X a Polish G-space with [x]_{G} a dense G_{δ}. Suppose y ∈ X has

\overline{[y]_{G}} = \overline{[x]_{G}}.
Choose (g_{i})_{i∈ℕ} and (h_{i})_{i∈ℕ} two sequences in G with
g_{i} · x → y
h_{i} · y → x.

It suffices to show that there are c_{0}, c_{1}, ..., c_{∞} ∈ G_{x} and g_{∞} ∈ G so that

g_{i}c_{i} → g_{∞}c_{∞}.

For each n ∈ ℕ let

V̂_{n} = {g ∈ G : d(g, 1) < 2^{−n}}

and let V_{n} ⊂ V̂_{n} be an open conjugation invariant neighborhood of the identity such that V_{n}^{−1} = V_{n} and (V_{n})^{2} ⊂ V̂_{n}. The map g ↦ g · x is open as a function from G to [x]_{G} by 7.12, so we can find open sets U_{n} containing x with V_{n} · x = U_{n} ∩ [x]_{G}. By speeding up the sequences (g_{i}) and (h_{i}) we may assume that for all n and for all i, j ≥ n

h_{i}g_{j} · x ∈ V_{n}.

Thus in particular

h_{j}g_{j} ∈ V_{j} · G_{x}

and so we may find c′_{j}, c′′_{j+1} ∈ G_{x} such that

h_{j}g_{j} ∈ V_{j}c′_{j}
h_{j}g_{j+1} ∈ V_{j}c′′_{j+1},
∴ (g_{j})^{−1}g_{j+1} ∈ (c′_{j})^{−1}(V_{j})^{−1}V_{j}c′′_{j+1},

and since V_{j} = (V_{j})^{−1} = c′_{j}(V_{j})^{−1}(c′_{j})^{−1}

(g_{j})^{−1}g_{j+1} ∈ (V_{j})^{2}(c′_{j})^{−1}c′′_{j+1}.

Let ĉ_{j} = ((c′_{j})^{−1}c′′_{j+1})^{−1}; this is in G_{x}, since the stabilizer of any point forms a subgroup of G. Rephrasing the above we have

(g_{j})^{−1}g_{j+1}ĉ_{j} ∈ (V_{j})^{2} ⊂ V̂_{j}

and thus by definition of the V̂_{j}

d((g_{j})^{−1}g_{j+1}ĉ_{j}, 1) < 2^{−j},

so by left invariance by d

d(g_{j+1}ĉ_{j}, g_{j}) < 2^{−j}.

Thus if we define c_{j} = ĉ_{j−1}ĉ_{j−2}...ĉ_{0} then for all j

d(g_{j+1}c_{j+1}, g_{j}c_{j}) < 2^{−j}

by the right invariance of d, and so for all l > j

d(g_{l}c_{l}, g_{j}c_{j}) < ∑_{i≥j} 2^{−i} = 2^{−j+1}

as required.

Corollary 7.24. (A) Let G be an abelian Polish group and X a Polish Gspace. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E_{G}^{X}.

(B) More generally let G be an invariantly metrizable Polish group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E_{G}^{X}.

Proof. (A) follows from (B) in light of 7.3. (B) follows from 7.23 by 7.11 and the definition of GE group.

Theorem 7.25. Any nilpotent Polish group is GE.

Proof. Supposing G to be a nilpotent Polish group amounts to asserting there is a finite sequence G = Z_{k} ▷ Z_{k−1} ▷ ....Z_{0} = {1} with each Z_{i} a normal subgroup of G such that Z_{i+1}/Z_{i} is the centralizer of G/Z_{i}: so that

Z_{i+1}/Z_{i} = {ḡ ∈ G/Z_{i} : ∀h̄ ∈ G/Z_{i}(ḡh̄ = h̄ḡ)}.

Let X be a Polish G space, x ∈ X. For each i let Z_{i}^{x} = \overline{⟨Z_{i}, G_{x}⟩} be the closure of the subgroup generated by Z_{i} and the stabilizer of x.

Claim(I): Each Z_{i}^{x}/G_{x} is closed in G/G_{x}.

Proof of Claim: It is a general fact that if G is a topological group and H_{0} < H_{1} < G are closed subgroups then the inclusion of H_{0} in H_{1} implies H_{1}/H_{0} is closed in G/H_{0}. (Claim(I)□)

Claim(II): Each Z_{i}^{x} is normal in Z_{i+1}^{x}.

Proof of Claim: Since the operation of conjugating one element by another is jointly continuous it suffices to show that for all g ∈ ⟨Z_{i+1}, G_{x}⟩ and h ∈ ⟨Z_{i}, G_{x}⟩

ghg^{−1} ∈ ⟨Z_{i}, G_{x}⟩.

So fix such g = z_{0}k_{0} and h = z_{1}k_{1}, with z_{0} ∈ Z_{i+1}, and z_{1} ∈ Z_{i}, and k_{0}, k_{1} ∈ G_{x}. Then

ghg^{−1} = z_{0}k_{0}z_{1}k_{1}(k_{0})^{−1}(z_{0})^{−1} = z_{0}(z_{0})^{−1}ẑ_{1}k_{0}k_{1}(k_{0})^{−1},

for some ẑ_{1} ∈ Z_{i} by the definition of Z_{i+1},

= ẑ_{1}k̂

for k̂ = k_{0}k_{1}(k_{0})^{−1} ∈ G_{x}. (Claim(II)□) Claim(III): Each Z_{i+1}^{x}/Z_{i}^{x} is abelian. Proof of Claim: It suffices to show that there is a dense abelian subgroup.

Letting

P r_{i+1} : Z_{i+1}^{x} → Z_{i+1}^{x}/Z_{i}^{x}
z ↦ zZ_{i}^{x}

be the projection map. ⟨Z_{i+1}, G_{x}⟩ is dense in Z_{i+1}^{x} and hence its image is dense in Z_{i+1}^{x}/Z_{i}^{x} under P r_{i+1}.

Fix g_{1}, g_{2} ∈ Z_{i+1} and k_{1}, k_{2} ∈ G_{x} with P r_{i+1}(g_{1}k_{1}) = ḡ_{1} and P r_{i+1}(g_{2}k_{2}) = ḡ_{2} both in Z_{i+1}^{x}/Z_{i}^{x}. Then by the definition of Z_{i+1}

ḡ_{2}ḡ_{1} = P r_{i+1}(g_{2}k_{2}g_{1}k_{1}) = P r_{i+1}(g_{1}g_{2}k_{1}k_{2}kϵ),

for some k = k_{2}^{−2}k_{1}^{−1}k_{2}k_{1} ∈ G_{x} and ϵ ∈ Z_{i}

= P r_{i+1}(g_{1}k_{1}g_{2}k_{2}kϵ′) = ḡ_{1}ḡ_{2}

some ϵ′ ∈ Z_{i}. (Claim(III)□)

Now suppose the orbit [x]_{G} is G_{δ} and that y ∈ X has

\overline{[y]_{G}} = \overline{[x]_{G}}.

Assume that for some l ∈ {0, 1, ..., k} there are g_{i} ∈ Z_{l}^{x} and h_{i} ∈ Z_{l+1}^{x} with

g_{i} · x → y

and

h_{i} · y → x,

and work towards showing that we can choose ĝ_{i} ∈ Z_{l−1}^{x′} and ĥ_{i} ∈ Z_{l}^{x′} with

ĝ_{i} · x′ → y

and

ĥ_{i} · y → x′

for some x′ ∈ [x]_{G}. (In the case that l = k take the h_{i}’s to be in Z_{k}^{x} and note that one of the steps below is superfluous.) This will prove that x and y are orbit equivalent, since by repeating the same argument over and over we at last obtain l = 0.

Claim(IV): There exists {z_{i} : i ∈ ℕ} ⊂ Z_{l} for which
z_{i} · x → y.

Proof of Claim: First choose ϵ_{i} → 1 in G and z_{i} ∈ Z_{l} and k_{i} ∈ G_{x} with g_{i} = ϵ_{i}z_{i}k_{i} by assumption of g_{i} ∈ Z_{l}^{x}. We may assume that there are open sets {O_{i} : i ∈ ℕ} around y and V_{i} at the identity of G thus:

{y} = ⋂_{i∈ℕ} O_{i};

V_{i} · O_{i} ⊂ O_{i−1};

g_{i} · x ∈ O_{i},

ϵ_{i} ∈ (V_{i})^{−1}.

Then

z_{i} · x = z_{i}k_{i} · x = ϵ_{i}^{−1}ϵ_{i}z_{i}k_{i} · x = ϵ_{i}^{−1}g_{i} · x ∈ O_{i−1}.
(Claim(IV)□)

Claim(V): There exist {ẑ_{i} : i ∈ ℕ} ⊂ Z_{l}^{x} for which

ẑ_{i} · y → x.

Proof of Claim: Fixing the z_{i} from claim(IV) and recalling h_{i} · y → x we may assume that

h_{i}z_{i} · x → x.

Then by 7.12 and claim (I)

[x]_{Z_{l+1}^{x}} =_{df} {g · x : g ∈ Z_{l+1}^{x}}

is G_{δ}; so one more application of 7.12 gives that

Z_{l+1}^{x}/G_{x} → [x]_{Z_{l+1}^{x}}
gG_{x} ↦ g · x

is open. Hence there are ϵ_{i} → 1 in Z_{l+1}^{x} such that

ϵ_{i}h_{i}z_{i}x = x.

Hence there are k_{i} ∈ G_{x} such that

ϵ_{i}h_{i}z_{i} = k_{i}.

Since Z_{l} ◁ G is a normal subgroup of G there are z̄_{i} ∈ Z_{l} such that

z̄_{i}ϵ_{i}h_{i} = k_{i}
∴ h_{i} = (ϵ_{i})^{−1}(z̄_{i})^{−1}k_{i}.

Since (ϵ_{i})^{−1} → 1 the argument from claim (IV) shows that setting ẑ_{i} = (z̄_{i})^{−1}k_{i} succeeds. (Claim(V)□)

Summing up: We now have [x]_{Z_{l}^{x}} ∈ G_{δ} and \overline{[x]_{Z_{l}^{x}}} = \overline{[y]_{Z_{l}^{x}}}, with witnesses {z_{i} : i ∈ ℕ} ⊂ Z_{l} and {ẑ_{i} : i ∈ ℕ} ⊂ Z_{l}^{x} for which

z_{i} · x → y,
ẑ_{i} · y → x.

As in the proof of 7.23 the openness of the map g ↦ g · x entails the existence of open neighborhoods V_{i} of 1 in G so that

ẑ_{i}z_{j} ∈ V_{i}G_{x}

for all j ≥ i,

V_{i+1} ⊂ V_{i},

and

{1} = ⋂_{i∈ℕ} V_{i}.

Thus if d^{∗} is an invariant two sided metric on Z_{l}^{x}/Z_{l−1}^{x} (which exists for any abelian group by 7.3 and is complete by 7.4) and P r_{l} : Z_{l}^{x} → Z_{l}^{x}/Z_{l−1}^{x} is the projection then we may assume that for all i and j ≥ i

d^{∗}(P r_{l}(ẑ_{i}z_{j}), 1) < 2^{−i−1}

and thus for all j, k ≥ i

d^{∗}(P r_{l}(ẑ_{i})P r_{l}(z_{j}), P r_{l}(ẑ_{i})P r_{l}(z_{k})) < 2^{−i}
∴ d^{∗}(P r_{l}(z_{j}), P r_{l}(z_{k})) < 2^{−i}

by right invariance. Hence we have a Cauchy sequence in this necessarily complete metric, and may therefore find z_{∞} ∈ Z_{l}^{x} with

P r_{l}(z_{i}) → P r_{l}(z_{∞}).

Thus there exist z_{i}^{∗} ∈ Z_{l−1}^{x} with

z_{i}z_{i}^{∗} → z_{∞}.

Thus choose δ_{i} ∈ Z_{l}^{x} with δ_{i} → 1 and at each i

δ_{i}z_{i}z_{i}^{∗} = z_{∞}.

Since Z_{l−1}^{x} is normal in Z_{l}^{x} we can find z̄_{i} ∈ Z_{l−1}^{x} with

z̄_{i}δ_{i}z_{i} = z_{∞}.

Let x′ = z_{∞} · x. Since z_{i} · x → y and δ_{i} → 1 we obtain

δ_{i}z_{i}x → y

and then

y ∈ \overline{[x′]_{Z_{l−1}^{x}}}

since

(z̄_{i})^{−1} · x′ = (z̄_{i})^{−1}z_{∞}x = (z̄_{i})^{−1}z̄_{i}δ_{i}z_{i} · x = δ_{i}z_{i} · x.
Claim(VI): G_{x} ⊂ Z_{l−1}^{x′}(=_{df} \overline{⟨Z_{l−1}, G_{x′}⟩}).

Proof of Claim: Fix h ∈ G_{x}. Then by z_{∞} ∈ Z_{l}^{x} there are z′_{i} ∈ Z_{l} and k_{i} ∈ G_{x} for which

(z′_{i})^{−1}(k_{i})^{−1}hk_{i}z′_{i} → (z_{∞})^{−1}hz_{∞}
∴ z_{∞}(z′_{i})^{−1}z_{∞}^{−1}z_{∞}k_{i}^{−1}z_{∞}^{−1}z_{∞}hz_{∞}^{−1}z_{∞}k_{i}z_{∞}^{−1}z_{∞}z′_{i}z_{∞}^{−1} → h

and so since G_{x′} = z_{∞}G_{x}z_{∞}^{−1}, for some k_{i}^{∗}, h^{∗} ∈ G_{x′}

z_{∞}(z′_{i})^{−1}z_{∞}^{−1}k_{i}^{∗}h^{∗}(k_{i}^{∗})^{−1}z_{∞}z′_{i}z_{∞}^{−1} → h;

and therefore by normality of Z_{l} there are z̃_{i} ∈ Z_{l} so that

z̃_{i}^{−1}k_{i}^{∗}h^{∗}(k_{i}^{∗})^{−1}z̃_{i} → h;

so by the definition of Z_{l}/Z_{l−1} as the center of G/Z_{l−1} there are ż_{i} ∈ Z_{l−1} with

ż_{i}k_{i}^{∗}h^{∗}(k_{i}^{∗})^{−1} → h.
(Claim(VI)□)

So: Z_{l−1}^{x} ⊂ Z_{l−1}^{x′} and Z_{l}^{x} ⊂ Z_{l}^{x′}.

We achieved

y ∈ \overline{[x′]_{Z_{l−1}^{x}}} ⊂ \overline{[x′]_{Z_{l−1}^{x′}}}

just prior to claim(VI), and

x′ ∈ \overline{[y]_{Z_{l}^{x}}} ⊂ \overline{[y]_{Z_{l}^{x′}}}

was claim(V).

Corollary 7.26. Let G be a nilpotent Polish group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{B} id(ℝ);

2. E_{0} ⊑_{c} E_{G}^{X}.

Proof. As at 7.24(B).

Exercise 7.27. (See [42].) Let ℤ^{ℤ} be given the product topology and group structure of pointwise multiplication. Define

ψ : ℤ → Hom(ℤ^{ℤ})

by

(ψ(l))(g_{k}) = g_{k−l}.

Let ℤ ⋉ ℤ^{ℤ} be the semi-direct product of ℤ and ℤ^{ℤ} along ψ, so that

(n_{0}, g^{0})(n_{1}, g^{1}) = (n_{0} + n_{1}, ((ψ(n_{1}))(g^{0}))g^{1}).

Let ℤ ∪ {∞} be the one point compactification of ℤ, and let ℤ ⋉ ℤ^{ℤ} act on (ℤ ∪ {∞})^{ℤ} by

((l, g⃗)(x⃗))_{k} = g_{k+l} + x_{k+l},

where n + ∞ is always ∞. Then (ℤ ∪ {∞})^{ℤ} is a Polish ℤ ⋉ ℤ^{ℤ}-space. (i) (..., 0, 0, 0, ....) and (.....0, 0, ∞, 0, 0, ...) both have dense orbits. (ii) The orbit of (....0, 0, 0, ...) is G_{δ}. (iii) ℤ ⋉ ℤ^{ℤ} is a solvable non-GE Polish group.

The group ℤ ⋉ ℤ^{ℤ} is (realizable as) a closed subgroup of S_{∞}, and thus by §6.1 and 7.18 all its orbit equivalence relations are classifiable by countable models. It seems that nothing specific can be said about the forms of the Glimm-Effros dichotomy satisfied by closed subgroups of S_{∞}.

7.3. Dynamic changes in topologies

For the purpose of §7.5 we work towards a theorem from [4] stating that given a Polish G-space X and a Borel set B we can expand the topology on X so that it is still a Polish G-space but B^{ΔV} now becomes open. The proof below is a little different and provides an exact evaluation of the damage inflicted on the original topology.

Definition 7.28. Let G be a Polish group and d a right invariant metric on G. We let L(G, d) be the space of all

f : G → ℝ^{≥0}(=_{df} {r ∈ ℝ : r ≥ 0})

such that for all g_{0}, g_{1} ∈ G

|f(g_{0}) − f(g_{1})| ≤ d(g_{0}, g_{1}).

We equip this with a topology whose subbasic open sets are of the form

{f ∈ L(G, d) : f(ĝ) < q_{1}}

and

{f ∈ L(G, d) : q_{0} < f(ĝ)}

where ĝ ∈ G and q_{0}, q_{1} ∈ ℚ^{+}(=_{df} {q ∈ ℚ : q > 0}). For g_{0} ∈ G and f_{0} ∈ L(G, d) define g_{0} · f_{0} by the requirement that for all g ∈ G

g_{0} · f_{0}(g) = f_{0}(gg_{0});

this is well defined since g_{0} · f_{0} ∈ L(G, d), by the right invariance of the metric; it is an action since for all g_{0}, g_{1}, g ∈ G and f ∈ L(G, d)

g_{1} · (g_{0} · f(g))) = g_{0} · f(gg_{1}) = f(gg_{1}g_{0}).

Lemma 7.29. L(G, d) is a Polish G-space.

Proof. (i) It is a Polish space: Fix {g_{i} : i ∈ ℕ} a countable dense subset of G. For f_{0}, f_{1} ∈ L(G, d) let

d_{L}(f_{0}, f_{1}) = ∑_{i∈ℕ} 2^{−i}|f_{0}(g_{i}) − f_{1}(g_{i})|/(1 + |f_{0}(g_{i}) − f_{1}(g_{i})|).

This metric generates the original topology, since for all ĝ ∈ G and f ∈ L(G, d) and rationals with q_{0} < f(ĝ) < q_{1} we can find some 1 > ϵ > 0 and i ∈ ℕ such that

d(g_{i}, ĝ) < ϵ
q_{0} + 2ϵ < f(ĝ) < q_{1} − 2ϵ.

Thus

f ∈ {f̂ ∈ L(G, d) : d_{L}(f, f̂) < ϵ2^{−i−1}} ⊂ {f̂ ∈ L(G, d) : q_{0} < f(ĝ) < q_{1}}.

(ii) The action is continuous: It suffices to check this for subbasic open sets. Fix g, h ∈ G and U including some subbasic open neighborhood

{f̂ ∈ L(G, d) : |f̂(h) − g · f(h)| < 2ϵ}

of some g · f ∈ L(G, d). Fix open

U_{0} = {f̂ ∈ L(G, d) : |f̂(hg) − f(hg)| < ϵ}

and V be an open neighborhood of g such that

∀ĝ ∈ V (d(hĝ, hg) < ϵ).

Then for all ĝ ∈ V and f̂ ∈ U_{0} we have

|ĝ · f̂(h) − g · f(h)| = |f̂(hĝ) − f(hg)|
≤ |f̂(hg) − f̂(hĝ)| + |f̂(hg) − f(hg)| < ϵ + ϵ.

Definition 7.30. Let X be a Polish G-space, d a right invariant metric on G. For O ⊂ X open define and x ∈ X define f_{x}^{O} : G → [0, 1] by

f_{x}^{O}(ĝ) = inf{d(g, ĝ)/(1 + d(g, ĝ)) : g · x ∈ O},

where this f_{x}^{O}(ĝ) is understood to be 1 when [x]_{G} ∩ O = ∅.

Lemma 7.31. The function

π_{O} : X → L(G, d)
x ↦ f_{x}^{O}

is G-map for all O ⊂ X and x ∈ X.

Proof. Fix x ∈ X and O ⊂ X and ĝ_{0}, ĝ_{1} ∈ G. Then
ĝ_{0} · f_{x}^{O}(ĝ_{1}) = inf{d(ĝ_{1}ĝ_{0}, g) : g · x ∈ O} = inf{d(ĝ_{1}, gĝ_{0}^{−1}) : g · x ∈ O}

by the right invariance of the metric; this in turn equals

= inf{d(ĝ_{1}, g) : g · (ĝ_{0} · x) ∈ O} =_{df} f_{ĝ_{0}·x}(ĝ_{1}).

The next definition and lemma formulates the precise extent the topology must be expanded to make the function π_{O} : x ↦ f_{x}^{O} continuous. The main point is that we need only add a collection of F_{σ} sets – each of the form B^{ΔV} =_{df} {x ∈ X : {g ∈ V : g · x ∈ B} is non-meager } for various closed sets in B ⊂ X. While that main, brutal point is all we quote at §7.5, later in chapter 9 we will need a precise determination of the collection of F_{σ} sets and the more subtle point that a presentation of a basis for the topology may be effectively produced: From bases for the topologies on X and G and a dense subgroup of G that all happen to be countable we can definably produce a countable basis for the new topology; the subtle point is that this can be done without first finding a countable enumeration of the basis for X. I will go ahead and present the notation B^{♯} and B^{+} that enables us to keep exact account of the new bases, but with the understanding that we will not draw on such a detailed analysis until 9.

Definition 7.32. Let B be a basis of X, {g_{i} : i ∈ ℕ} = G_{0} a countable dense subgroup of G, and B_{0} be a basis for G including all sets of the form {g ∈ G : d(g, g_{i}) < q} for i ∈ ℕ and q ∈ ℚ. Let C be a closed set in X. Then set B^{♯}(B, B_{0}, C) to consist of all sets of the form

(U ∩ C^{∗V_{0}} ∩ C^{∗V_{1}} ∩ ... ∩ C^{∗V_{k}})^{ΔW} ,

where V_{0}, V_{1}, ..., V_{k}, W in B_{0}.

For C a collection of closed sets in X, let B^{+}(B, B_{0}, C) be the smallest collection of sets including B and {(X \ C) : C ∈ C}, closed under finite intersections and unions, and such that for all U ∈ B^{+}(B, B_{0}, C)

B^{♯}(B^{+}(B, B_{0}, C), B_{0}, U \ X) ⊂ B^{+}(B, B_{0}, C).

Note that this last inclusion is nothing other than the requirement that for all U_{0} ∈ B^{+}(B, B_{0}, C) and V_{0}, V_{1}, ..., V_{k}, W in B_{0} we have

(U_{0} ∩ C^{∗V_{0}} ∩ C^{∗V_{1}} ∩ ... ∩ C^{∗V_{k}})^{ΔW} ∈ B^{+}(B, B_{0}, C)

for C = X \ U.

Lemma 7.33. Let B and B_{0} be be bases of X and G as in 7.32. Let O ⊂ X be open and C = X \ O be the closed set that forms its complement. Then for every subbasic Û ⊂ L(G, d) the pullback π_{O}^{−1}[Û] is a countable union of elements in B^{♯}(B, B_{0}, C).

Proof. The pullbacks of subbasic open sets of the form

{f ∈ L(G, d) : f(g) < q}

are already open in X for q > 0 and g ∈ G. On the other hand for subbasic sets of the form

{f_{x}^{O}(g) > q}

it suffices as in the proof of 7.29(i) to consider the case that q ∈ ℚ and g = g_{i} for some i ∈ ℕ. But then if we have some fixed x_{0} ∈ X with

f_{x_{0}}^{O}(g_{i}) > q

then there will be some rational q̄ > q such that

f_{x_{0}}^{O}(g_{i}) ≥ q̄

which amounts to saying that

∀g ∈ G(d(g, g_{i}) < q̄ ⇒ g · x_{0} ∈ C);

by right invariance of d this amounts to asserting that for V = {ĝ ∈ G : d(ĝ, 1) < q̄}

∀g ∈ V g_{i}(gg_{i} · x_{0} ∈ C);

equivalently:

x_{0} ∈ C^{∗V g_{i}}.

In short we have seen that for any x_{0} in the set

π^{−1}[{f_{x_{0}}^{O}(g_{i}) > q}]

we may find a set C^{∗V g_{i}} ∈ B^{♯}(B, B_{0}, C) with

x_{0} ∈ C^{∗V g_{i}} ⊂ π^{−1}[{f_{x_{0}}^{O}(g_{i}) > q}].

Lemma 6.9 implies that all the sets in B^{♯}(B, B_{0}, C) are certainly no worse than F_{σ}. Thus the open sets in the topology generated by B^{♯}(B, B_{0}, C) are all F_{σ} with respect to the original topology on X.

Definition 7.34. Let X, G, d be as above, O ⊂ X open. Let X̂(O) = {(x, f) ∈ X × L(G, d) : f = f_{x}^{O}}.

Lemma 7.35. Let X, G, d be as above, O ⊂ X open. Then X̂(O) is a G_{δ} subset of X × L(G, d) (in the product topology).

Proof. Fix {g_{i} : i ∈ ℕ} a dense subgroup of G. For each i ∈ ℕ and q ∈ ℚ the set

{(x, f) ∈ X × L(G, d) : f(g_{i}) > q > f_{O}^{x}(g_{i})}

is open by definition and the set

{(x, f) ∈ X × L(G, d) : f(g_{i}) < q < f_{O}^{x}(g_{i})}

is F_{σ} by 7.33.

Lemma 7.36. Let X be a set with τ a Polish topology on X. Suppose that G is a Polish group acting on X, (X, τ) is a Polish G-space, and C ⊂ X is closed with respect to τ. Then there is a new topology τ̂ so that:

1. τ ⊂ τ̂;

2. (X, τ̂) is a Polish G-space;

3. for each W ⊂ G open

C^{ΔW} ∈ τ̂;

4. every open set in (X, τ̂) is F_{σ} in (X, τ); moreover if B and B_{0} are bases as in 7.32 we may also obtain:

1. τ̂ has B^{♯}(B, B_{0}, C) as a basis.

Proof. Fix {g_{i} : i ∈ ℕ} = G_{0} a dense subgroup of G and B a basis of (X, τ). Let O = X \ C and define ( id, π_{O}) : X → X̂(O) by

x ↦ (x, f_{O}^{x}).

Let τ̂ be the topology generated by the pullback of open sets in X̂(O).

It is then immediate from the definition of the product topology on X ×L(G, d) that τ̂ ⊃ τ. 2 follows from 7.31 and 7.35 and the general fact from 2.3 that G_{δ} subspaces of Polish spaces are Polish. As for 3, note that for x ∈ X and W ⊂ G open

x ∈ C^{ΔW} ⇔ ∃g ∈ W (f_{O}^{x}(g) > 0).

4 follows from 7.33, since all the sets in B^{♯}(B, B_{0}, C) are F_{σ} in X.

For the further statement at 5 we need to do a little more in the defining of τ̂. Let O⃗ = (O_{i})_{i∈ℕ} enumerate all the open sets arising as complements of sets of the form C^{∗V_{0}} ∩ C^{∗V_{1}} ∩ ... ∩ C^{∗V_{k}} where each V_{i} ∈ B_{0} and then by analogy with above define

( id, π_{O_{0}}, π_{O_{1}}, ...) : X → X̂(O⃗)
x ↦ (x, f_{O_{0}}^{x}, f_{O_{1}}^{x}, ...)

and take the pullback of the topology along ( id, π_{O_{0}}, π_{O_{1}}, ...). We obtain that this is a Polish topology on X as in the proof of 2 above. By the argument for 3 above we have enforced that B^{♯}(B, B_{0}, C) is included in this new topology. Finally to see that the new topology has B^{♯}(B, B_{0}, C) as a basis note that

(C^{∗V_{0}} ∩ C^{∗V_{1}} ∩ ... ∩ C^{∗V_{k}})^{∗W} = C^{∗V_{0}W} ∩ C^{∗V_{1}W} ∩ ... ∩ C^{∗V_{k}W} ,

and so we can apply 7.33.

Corollary 7.37. For G, X, B, B_{0} as above, C a countable collection of closed sets, the space X in the topology generated by B^{+}(B, B_{0}, C) remains a Polish Gspace.

Proof. By applying 7.36 countably many times we find sequence of Polish topologies τ_{0} = τ ⊂ τ_{1} ⊂ ...τ_{i}... with countable bases B_{0}^{∗} = B ⊂ B_{1}^{∗} ⊂ ...B_{i}^{∗}..., each closed under finite intersections and unions, each included in B^{+}(B, B_{0}, C), such that for each i ∈ ℕ and U ∈ B_{i}^{∗}, there is some j > i with

B_{j+1}^{∗} ⊃ B^{♯}(B_{j}^{∗}, B_{0}, (X \ U))

and for each C ∈ C there is some k with

B_{k+1}^{∗} ⊃ B_{k}^{∗} ∪ {U \ C : U ∈ B_{k}^{∗}}.

By 2.3 we can finish with τ̂ the Polish topology generated by

B̂ = ⋃_{i∈ℕ} B_{i}^{∗}.

We do then have that for all U_{0}, U ∈ B_{i}^{∗}, and V_{0}, V_{1}, ..., V_{k}, W ∈ B_{0}, if we let C = X \ U then there is some j > i with

(U_{0} ∩ C^{∗V_{0}} ∩ C^{∗V_{1}} ∩ ... ∩ C^{∗V_{k}})^{ΔW} ∈ B_{j}^{∗}.

Hence B^{+}(B, B_{0}, C) ⊂ B̂. We maintained the reverse inclusion at every step of the construction, and thus we have B^{+}(B, B_{0}, C) = B̂, as required.

Theorem 7.38. (Becker, Kechris) Let (X, τ) be a Polish G-space and B ⊂ X Borel. Then there is a topology τ̂ such that:

1. τ ⊂ τ̂;

2. (X, τ̂) is a Polish G-space;

3. for each W ⊂ G open

B^{ΔW} ∈ τ̂;

4. every open set in τ̂ is Borel in τ.

Proof. Let D be the collection of B for which there is some τ̂ as in the conclusion of the theorem. It includes all open sets and it is closed under countable unions by 2.3(iii). As for complementation note that

(X \ B)^{ΔV}

is equal to the union over basic open W_{0} and W_{1} with W_{0}W_{1} ⊂ V of the sets

((X \ B)^{∗W_{0}})^{ΔW_{1}}.

Thus if we can find a topology (τ)^{♯} in which each

B^{ΔW_{0}}

becomes open then we may find a topology (τ)^{♯♯} in which the Δ Vaught transform of the complement

((X \ B)^{∗W_{0}})^{ΔW_{1}}

become open by 7.36.

Exercise 7.39. ([32]) For B ∈ Σ ∼_{α}^{0}(X, τ) at we may tighten part 4 of 7.38 4^{∗}. every open set in τ̂ is Σ ∼_{α}^{0}(X, τ).

7.4. Products of locally compact groups

Theorem 7.40. Let G = ∏_{i∈ℕ} G_{i} in the product topology and group structure with each G_{i} a locally compact Polish group. Then G is a GE group.

Proof. Let X be a Polish G-space and x, y ∈ X with

[x]_{G} ∈ G_{δ}
\overline{[x]_{G}} = \overline{[y]_{G}} = X.

Notation 7.41. Let

H_{l} = {g ∈ G : ∀n < l(g(n) = 1)}

be the closed normal subgroup of G consisting of all elements that take the identity on coordinates below l. For x′ ∈ [x]_{G} let

H_{l}^{x′} = \overline{⟨H_{l}, G_{x′}⟩}

(where as usual G_{x′} indicates {g ∈ G : g · x′ = x′} and the overline indicates closure). For

g ∈ ∏_{i≤k} G_{i}

and

h ∈ ∏_{i>k} G_{i}

let g^{⌢}h ∈ G be defined by (g^{⌢}h)(n) = g(n) for n ≤ k but (g^{⌢}h)(n) = h(n) for n > k.

The form of the proof is to inductively obtain sequences of

δ̂_{l} ∈ ∏_{i<l} G_{i}

h_{l}^{∗} ∈ G

x_{l} = h_{l}^{∗} · x_{0} = h_{l}^{∗} · x

such that for d^{k} some fixed right invariant metric on G_{k} and l > k

d^{k}(δ̂_{l}(k), 1) < 2^{−l}

and such that

h_{l+1}^{∗}(h_{l}^{∗})^{−1}|_{l+1} = δ̂_{l};

this will also be done so

y ∈ \overline{[x_{l}]_{H_{l}^{x_{l}}}} ,
d(x_{l}, y) < 2^{−l}.

If we achieve all this the theorem is proved: Since any right invariant metric on a locally compact group is complete so it follows that there exists h_{∞} with

h_{l}^{∗}(i) → h_{∞}(i)

at each i, and thus

h_{l}^{∗} → h_{∞}

and thus

h_{∞} · x = y.

So let us just concentrate on showing we can complete the construction of the

x_{0}, x_{1}, ..., δ̂_{0}, δ̂_{1}, ..., h_{0}^{∗}, h_{1}^{∗}, ...

Assume inductively that we have x_{l} ∈ [x]_{G} and

y ∈ \overline{[x_{l}]_{H_{l}^{x_{l}}}} .

Claim(I): x_{l} ∈ \overline{[y]_{H_{l}^{x_{l}}}}.

Proof of claim: Choose h_{i} ∈ G

h_{i} · y → x_{l}

and ĝ_{i} ∈ H_{l}^{x_{l}} with

ĝ_{i} · x_{l} → y.

Note that we may choose g_{i} ∈ H_{l} and

ϵ_{i} → 1_{G},

and

k_{i} ∈ G_{x_{l}}

so that

ĝ_{i} = ϵ_{i}g_{i}k_{i}.

Subclaim: g_{i} · x_{l} → y.

Proof of subclaim: Then for any open O containing y we may choose open O_{0} ⊂ O containing y and V containing 1_{G} so that

V · O_{0} ⊂ O.

Then for for i large enough that ϵ_{i}^{−1} ∈ V and ĝ_{i} · x_{l} ∈ O_{0}

g_{i} · x_{l} = ϵ_{i}^{−1}ϵ_{i}g_{i}k_{i} · x_{l} = ϵ_{i}^{−1}ĝ_{i} · x_{l} ∈ O.

(Subclaim□)

We may assume

h_{i}g_{i} · x_{l} → x_{l}.

By 7.12 the map

G/G_{x_{l}} → G · x_{l}
gG_{x_{l}} ↦ g · x_{l}

is a homeomorphism, so there are k̂_{i} ∈ G_{x_{l}} and

δ_{i} → 1_{G}

such that

δ_{i}h_{i}g_{i} = k̂_{i}.
∴ δ_{i}h_{i} = k̂_{i}(g_{i})^{−1} = ḡ_{i}k̂_{i}

some ḡ_{i} ∈ H_{l} by normality of H_{l}.

∴ δ_{i}^{−1}ḡ_{i}k̂_{i} · y = h_{i} · y → x_{l}
∴ ḡ_{i}k̂_{i} · y → x_{l},

by the argument of the subclaim. (Claim(I)□)

Let X_{l} = \overline{[x_{l}]_{H_{l}^{x_{l}}}}, the closure of the orbit x_{l} of under H_{l}^{x_{l}}. From claim(I), [y_{l}]_{H_{l}^{x_{l}}} is dense in X_{l}.

Since H_{l}^{x_{l}}/G_{x_{l}} is closed in G/G_{x_{l}} it follows by 7.12 that [x_{l}]_{H_{l}^{x_{l}}} is relatively closed in [x_{l}]_{G}, and hence it is a G_{δ} set; so the map

H_{l}^{x_{l}}/G_{x_{l}}

hG_{x_{l}} ↦ h · x_{l}

is a homeomorphism by 7.12 again. Thus letting

V ⊂ ∏_{k≤l} G_{k}

be an open set with compact closure and letting

V_{0} = {h ∈ H_{l}^{x_{l}} : h|_{l+1} ∈ V }

then

\overline{V_{0} · x_{l}} ∩ [y]_{H_{l}^{x_{l}}} ≠ ∅.

Fix y_{1} ∈ \overline{V_{0} · x_{l}} ∩ [y]_{H_{l}^{x_{l}}}.

Claim(II): There exists x′ ∈ [x_{l}]_{H_{l}^{x_{l}}} with y_{1} ∈ \overline{H_{l+1} · x′}.

Proof of claim: By the assumption on y_{1} choose ḡ_{i} = h_{i}^{⌢}g_{i} ∈ V_{0}, so that h_{i} ∈ V and g_{i} ∈ ∏_{k>l} G_{k},

ḡ_{i} · x_{l} → y_{1}.

By the compactness of \overline{V} we may assume that there is some h ∈ V̄ such that

h_{i} → h.

Let 1 ∈ ∏_{k>l} G_{k} be the group element that is constantly the identity at every coordinate and let 1^{l} be the identity element in ∏_{k≤l} G_{k}. Each h_{i}^{⌢}1 is in H_{l}^{x_{l}} since h_{i}g_{i} ∈ H_{l+1} ⊂ H_{l}^{x_{l}}. Since H_{l}^{x_{l}} is closed

h^{⌢}1 ∈ H_{l}^{x_{l}}.

Since h_{i} → h there are
δ_{i} → 1_{G}

such that

δ_{i}(h^{⌢}1)h_{i}^{⌢}1.

Hence

ḡ_{i} = δ_{i}(h^{⌢}1)((1^{l})^{⌢}g_{i}) = δ_{i}((1^{l})^{⌢}g_{i})(h^{⌢}1)

and therefore as in the subclaim from claim(I),

((1^{l})^{⌢}g_{i})(h^{⌢}1) · x_{l} → y_{1},

and so (h^{⌢}1) · x_{l} = x′ is as required. (Claim(II)□)

By the assumption on y_{1} choose g ∈ H_{l}^{x_{l}} with g · y = y_{1}. Then for x_{l+1} = g^{−1} · x′ ∈ [x_{l}]_{H_{l}^{x_{l}}} we have

y = g^{−1}y_{1} ∈ g^{−1}\overline{H_{l+1} · x′} = \overline{g^{−1}H_{l+1} · x′}
= \overline{g^{−1}H_{l+1}gg^{−1} · x′} = \overline{H_{l+1}g^{−1} · x′},

since H_{l+1} is normal,

= \overline{H_{l+1}x_{l+1}},

by definition of x_{l+1}.

So we have x_{l+1} ∈ [x_{l}]_{H_{l}^{x_{l}}} with y ∈ \overline{H_{l+1} · x_{l+1}} and so certainly

y ∈ \overline{H_{l+1}^{x_{l+1}} · x_{l+1}}.

By possibly replacing x_{l+1} by some other point in H_{l+1} · x_{l+1} we may assume that

d(x_{l+1}, y) < 2^{−l−1}.

By x_{l+1} ∈ [x_{l}]_{H_{l}^{x_{l}}} fix δ̄_{i} ∈ ∏_{k<l} G_{k} and ĥ_{i} ∈ ∏_{k≥l} G_{k} with

δ̄_{i} → 1
x_{l+1} = (δ̄_{i}^{⌢}ĥ_{i}) · x_{l}.

Then for i large enough that δ̄_{i} close enough to the identity that for all k ≤ l

d^{k}(δ̄_{i}, 1_{G_{k}}) < 2^{−l−1}

we can take

δ̂_{l} = \overline{δ_{i}}
h_{l+1}^{∗} = (δ̄_{i}^{⌢}ĥ_{i})h_{l}^{∗}

and continue with the construction.

Theorem 7.42. Let G = ∏_{i∈ℕ} G_{i} be a product of locally compact Polish groups. Then G has no turbulent actions.

Proof. Let X be a Polish G-space. Assuming every orbit is meager we will attempt to head off part (iii) in the definition of turbulence.

For each n let d^{n} be a complete right invariant metric on

G^{n} =_{df} ∏_{i≤n} G_{n}.

As in 7.28 let L(G^{n}, d^{n}) be the Polish G^{n}-space of function f : G^{n} → ℝ^{≥0} with |f(h) − f(h′)| always bounded by d^{n}(h, h′). Let

X_{n} = ∏_{ℕ} L(G^{n}, d^{n})

be the product of L(G^{n}, d^{n}) with itself ℵ_{0} many times, viewed as a Polish G^{n}-space in the product topology and G^{n}-action.

Fixing a countable basis {U_{m} : m ∈ ℕ} for X define for each x ∈ X and n, m ∈ ℕ the function

f_{⟨n,m⟩}^{x} : G^{n} → ℝ
g ↦ inf{d^{n}(ĝ, g) : ∃h⃗ ∈ G(ĝ ⊂ h⃗, ĝ · x ∈ U_{m})}.

We then define

ϕ_{n} : X ↦ X_{n}
x ↦ (f_{⟨n,m⟩}^{x})_{m∈ℕ}.

Claim(I): For all g ∈ G^{n} and h⃗ ∈ G with g ⊂ h⃗ and x ∈ X

ϕ_{n}(h⃗ · x) = g · ϕ_{n}(x).

Proof of claim: As in the proof of 7.31. (Claim(I)□) By Borel or even Σ ∼_{1}^{1} sets having Baire property we may find a comeager C ⊂ X

on which each ϕ_{n} is continuous. Choose x ∈ C so that ∀^{∗}g ∈ G(g · x ∈ C). We will show that x fails (iii) from the definition of turbulence.

Since [x]_{G} is meager, we may find closed nowhere dense B_{i} ⊂ X with

[x]_{G} ⊂ ⋃_{i∈ℕ} B_{i}.

Since the map from G to X given by g ↦ g ·x is continuous we can find some h ∈ G and basic open neighborhood V ⊂ G of 1 so that

hV · x ⊂ B_{i},
∴ V · x ⊂ h^{−1}B_{i}.

Note that h^{−1}B_{i} is still closed nowhere dense.

Since V is basic open it is supported on only finitely many coordinates of G. For notational simplicity assume that there is some open W ⊂ G_{0} such that

V = {g⃗ ∈ G : g(0) ∈ W }.

Refining we can find an open neighborhood W_{0} ⊂ W of the identity in G_{0} such that

(W_{0})^{−1} = W_{0}
(W_{0})^{5} ⊂ W

and then let

V_{0} = {g⃗ ∈ G : g(0) ∈ W_{0}}.

Claim(II): There is some open O ⊂ X_{0} containing φ_{0}(x) and such that for all g ∈ (W_{0})^{4}

g · ϕ_{0}(x) ∈ O

implies

g · ϕ_{0}(x) ∈ W_{0} · ϕ_{0}(x).

Proof of claim: (An equivalent point is made in the course of [49].) Otherwise let O_{n} be open sets around ϕ_{0}(x) with O_{n+1} ⊂ O_{n} and

{ϕ_{0}(x)} = ⋂_{n∈ℕ} O_{n}.

Then choose g_{i} ∈ (W_{0})^{4}

g_{i} · ϕ_{0}(x) ∈ O_{i}
∴ g_{i} · ϕ_{0}(x) → ϕ_{0}(x),

but

g_{i} · ϕ_{0}(x) ∉ W_{0} · ϕ_{0}(x).

By compactness of \overline{W} there is some g_{∞} ∈ G_{0} with

g_{i} → g_{∞}
∴ g_{∞} · ϕ_{0}(x) = ϕ_{0}(x).

So procure

δ_{i} → 1

with

δ_{i}g_{i} = g_{∞}.

Then at i large enough that (δ_{i})^{−1} ∈ W_{0} we have

g_{i} · ϕ_{0}(x) = (δ_{i})^{−1}g_{∞} · ϕ_{0}(x) ∈ W_{0} · ϕ_{0}(x).
(Claim(II)□)

For this O and our comeager C on which all our functions are continuous, choose open O′ ⊂ X so that ϕ_{0}^{−1}[O] ∩ C = O′ ∩ C.

Claim(III): For all z ∈ O(x, O′, V_{0})

ϕ_{0}(z) ∈ (W_{0})^{2} · ϕ_{0}(x).

Proof of claim. Let g_{i} ∈ V_{0} and x_{i} ∈ O′ be the witnesses to this, so that x_{0} = x, x_{i+1} = g_{i} · x_{i}, and z = x_{k}. We need to show by induction on i that

ϕ_{0}(x_{i}) ∈ (W_{0})^{2} · ϕ_{0}(x).

Given its truth at i, we may discharge the assumption ∀^{∗}g ∈ G(g · x ∈ C) to find g ∈ V_{0} such that

g · x_{i+1} = gg_{i} · x_{i} ∈ O′ ∩ C
∴ ϕ_{0}(g · x_{i+1}) ∈ (W_{0})^{4} · ϕ_{0}(x) ∩ O
∴ ϕ_{0}(g · x_{i+1}) ∈ W_{0} · ϕ_{0}(x)
∴ ϕ_{0}(x_{i+1}) ∈ (W_{0})^{2} · ϕ_{0}(x)

by claims(I) and (II). (Claim(III)□)

Let

B = {z ∈ X : ∀g ∈ V_{0}(g · z ∈ h^{−1}B_{i}}.

This contains x and is closed. By 2.47 we have a comeager set of y for which

∀^{∗}g ∈ G(g · y ∈ (X \ h^{−1}B_{i}));

for such y

∀g ∈ G(g · y ∉ B).

It suffices to show that for some U′ and V′ the closure of the local U′-V′−orbit of x does not include any point in [y]_{G}.

Now let y′ be in [y]_{G}, and assume for a contradiction that y′ ∈ \overline{O(x, O′, V_{0})}. Since y′ is not in the closed set B we may choose basic open U_{m} containing y′ with U_{m} and B disjoint. Then for any z ∈ O(x, O′, V_{0}) with z ∈ U_{m} it follows that

f_{⟨0,m⟩}^{z}(1_{G_{0}}) = 0
∴ ∃g ∈ (V_{0})^{2}(f_{⟨0,m⟩}^{g·x}(1_{G_{0}}) = 0)

by claim (III). Thus by definition of this function we may find

ϵ_{i} → 1_{G_{0}}

h_{i} ∈ ∏_{n≥1} G_{n}

(ϵ_{i}^{⌢}h_{i})g · x ∈ U_{m},

and so for great enough i

ϵ_{i} ∈ W_{0}

and thus

(ϵ_{i}^{⌢}h_{i})g ∈ (V_{0})^{3},

with a contradiction to (V_{0})^{3} · x ⊂ B and B ∩ U_{m} = ∅.

Corollary 7.43. Let G = ∏ G_{n} be a countable product of locally compact groups and X a Polish G-space. Then there is a countable language L and Polish subspace Y ⊂ Mod(L) and Borel

θ : X → Mod(L)
ρ : B → X

such that 1. θ witnesses E_{G}^{X} ≤_{B} E_{S_{∞}}^{Mod(L)};

2. ρ witnesses E_{S_{∞}}^{Y} ≤_{B} E_{G}^{X};

3. θ and ρ are inverses up to orbit equivalence, in that for all x ∈ X

ρ ∘ θ(x)E_{G}^{X}x.

Proof. By 7.42, 7.40, and 6.40.

Exercise 7.44. Let G be a locally compact Polish group and X a Polish Gspace. Show that for all x ∈ X

γ(x) ≤ 1,

and thus since locally compact groups are GE

E_{G}^{X} ≤_{B} F_{2}.

(But [49] has a sharper result: we can find a countable discrete group H and a Polish H-space Y with E_{G}^{X} ≤_{B} E_{H}^{Y}.)

Exercise 7.45. Given a countable partially ordered set (I, ≤), locally compact Polish groups (G_{i})_{i∈I} and continuous onto homomorphic maps ϕ_{ij} : G_{i} → G_{j} for i ≥ j with the commutativity condition that i ≥ j ≥ k implies ϕ_{ik} = ϕ_{jk} ∘ ϕ_{ij}, we define the direct limit of (G_{i}, ϕ_{i,j})_{i≤j∈I} to be the “least” G for which there are continuous onto homomorphisms ϕ_{i} : G → G_{i} so that for all i ≥ j we have ϕ_{ij} ∘ ϕ_{i} = ϕ_{j} – least in the sense that given any other H with ψ_{i} : H ↠ G_{i} and each ϕ_{ij} ∘ ψ_{i} = ψ_{j} we have π : H ↠ G with each ψ_{i} = ϕ_{i} ∘ π. (Compare §2 [72].)

Then G is a closed subgroup of

∏_{i∈I} G_{i},

and thus 7.43 and 7.18 all its orbit equivalence relations admit classification by countable models.

7.5. CLI groups have the weak Glimm-Effros property

Definition 7.46. A Polish group G is said to have the weak Glimm-Effros property if whenever X is a Polish G-space one of the following must hold:

(I) E_{G}^{X} ≤_{B} id(ℝ);

(II) E_{0} ⊑_{c} E_{G}^{X}.

Definition 7.47. A Polish group is said to be CLI if it admits a complete left invariant metric.

We could similarly define a group to be RLI if it admits a complete right invariant complete metric, but this is an equivalent notion. Given d_{l} a left invariant complete metric we obtain a right invariant complete metric with

d_{r}(g, h) = d_{l}(g^{−1}, h^{−1}).

An essential step in the proof of Howard Becker’s theorem at 7.49 below is to prepare the topology of the Polish G-space. This is expressed at 7.48. Ideally we might like to have a basis for the topology that is closed under all Boolean combinations and Vaught transforms, but this is beyond hope. 7.48(3) instead negotiates an approximation: we will at least have the Δ Vaught transforms of all Boolean combinations of basic open sets.

Lemma 7.48. Let G be a Polish group, (X, τ) a Polish G-space, and B a countable basis for (X, τ) and B_{0} a countable basis for G. Then there is a topology τ̂ ⊃ τ with basis B̂ such that:

1. τ ⊂ τ̂ and B ⊂ B̂;

2. (X, τ̂) is a Polish G-space;

3. for all W_{0}, W_{1} ⊂ G basic open and U, U′ ∈ B̂

((X \ U)^{∗W_{0}} ∩ U′)^{ΔW_{1}} ∈ B̂;

4. every open set in τ̂ is Borel in τ.

Proof. B^{+}(B, B_{0}, ∅) from 7.32 suffices in light of 7.37.

Lemma 7.49. Let G be a Polish group and X a Polish G-space. Let [x]_{G} be a G_{δ} orbit. Then for any open neighborhood W of G and x_{0} ∈ [x]_{G} we may find some open U′ ⊂ X such that:

1. x_{0} ∈ U′;

2. for any other x_{1} ∈ U′ ∩ [x]_{G} and any other open U with

[x]_{G} ∩ U ∩ U′ ≠ ∅

there exists g ∈ W with

g · x_{1} ∈ U.

Proof. Choose V ⊂ W an open neighborhood of the identity with

V^{2} ⊂ W,
V^{−1} = V.

Then by 7.12 we have the function

G → [x]_{G}
g ↦ g · x_{0}

to be open. Thus in particular we have that V · x_{0} is a relatively open set in [x]_{G}, and thus we may find open U′ ⊂ X with U′ ∩ [x]_{G} = V · x_{0}.

Now for any other x_{1} ∈ U′ ∩ [x]_{G} we may find some g_{1} ∈ V with

g_{1} · x_{0} = x_{1}.

And if U is an open set with non-empty intersection with

U′ ∩ [x]_{G} = V · x_{0}

then we may find some g_{2} ∈ V with

g_{2} · x_{0} ∈ U.

Then

g_{2}g_{1}^{−1} · x_{1} ∈ U′,

as required.

Lemma 7.50. Let G be a Polish group and (X, τ) a Polish G-space. Let B_{0} be basis for G and B a basis for (X, τ). Suppose that for all W_{0}, W_{1} ∈ B_{0} and U, U′ ∈ ∩B we have that the set

((X \ U)^{∗W_{0}} ∩ U′)^{ΔW_{1}}

is τ-open.

Suppose [x]_{G} is a dense G_{δ} orbit. Then for any V an open neighborhood of the identity in G there is open U′

containing x such that for all open U ⊂ X and y ∈ U′

U ∩ U′ ≠ ∅ ⇒ (y ∈ U^{ΔV}).

Proof. Choose W ∈ B_{0} a basic open neighborhood of the identity with W^{2} ⊂ V. Choose U′ as in 7.49 for W, and after possibly going to an open subset we may assume U′ ∈ B. Suppose U ⊂ X is basic open with U ∩ U′ ≠ ∅; we wish to show that for any y ∈ U′ we have some g ∈ V with g · y ∈ U.

But if not, then we must have for this y

∀g ∈ W ∀h ∈ W (h · (g · y) ∈ (X \ U)),
∴ ∀g ∈ W (g · y ∈ (X \ U)^{∗W}).

On the other hand, clearly

∃g ∈ W (g · y ∈ U′).

Thus we have

y ∈ ((X \ U)^{∗W} ∩ U′)^{ΔW} .

Since this is an open set in our topology on X and since [x]_{G} is dense, we may find some

x_{1} ∈ ((X \ U)^{∗W} ∩ U′)^{ΔW} ∩ [x]_{G}.

Then in particular we may choose g_{1} ∈ W with

g_{1} · x_{1} ∈ (X \ U)^{∗W} ∩ U′.

Since U is open this implies that for all g_{2} ∈ W

g_{2} · (g_{1} · x) ∉ U,

which crashes into a contradiction with the assumption on U′ and the facts that

g_{1} · x ∈ [x]_{G}

and

U ∩ U′ ≠ ∅.

Corollary 7.51. Let G be a Polish group with basis B_{0} and compatible metric d and (X, τ) a Polish G-space with basis B. Suppose B has the closure property indicated at 7.48(3): for all W_{0}, W_{1} ⊂ G basic open and U, U′ ∈ B̂

((X \ U)^{∗W_{0}} ∩ U′)^{ΔW_{1}} ∈ B̂.

Suppose [x]_{G} is G_{δ}.

Then there is a sequence of open sets

U_{n+1} ⊂ U_{n}

with

{x} = ⋂_{n∈ℕ} U_{n},

such that for all n and for all y ∈ U_{n} with

\overline{[y]_{H}} = \overline{[x]_{H}}

and for all U ⊂ X open with [x]_{G} ∩ U_{n} ∩ U ≠ ∅ there exists g ∈ G such that

d(1, g) < 2^{−n}
g · y ∈ U.

Proof. Apply 7.50 to

{g ∈ G : d(1, g) < 2^{−n}}

inside the Polish space

\{y ∈ X : \overline{[y]_{G}} ⊂ \overline{[x]_{G}}\}.

Theorem 7.52. (Becker) Every CLI group has the weak Glimm-Effros property.

Proof. Let G be a CLI group and X a Polish G-space. Let d_{r} be a complete right invariant complete metric on G. Let B_{0} be a countable basis for G.

Starting with lemma 7.48 we may assume that X has a countable basis B such that for all U, U′ ∈ B and W_{0}, W_{1} ∈ B_{0} the set ((X \ U)^{∗W_{0}} ∩ U′)^{ΔW_{1}} ∈ B. Assume that E_{0} ⊑_{c} E_{G}^{X} does not hold. By 7.11 it suffices to show that for all y ∈ X we have

[y]_{G} ∈ G_{δ}.

By the assumption that E_{0} ⋢_{c} E_{G}^{X} and 7.8 we can find for all y ∈ X some x ∈ X with

\overline{[x]_{G}} = \overline{[y]_{G}},
[x]_{G} ∈ G_{δ}.

Let us assume then that [x]_{G} ∈ G_{δ} and \overline{[x]_{G}} = \overline{[y]_{G}} and attempt to show xE_{G}^{X}y. Following 7.51 we may find a sequence of U_{n} ∈ B so that

⋂ U_{n} = {x}

at each n and for each y′ ∈ [y]_{G} ∩ U_{n} and U ⊂ X open with U_{n} ∩ U ≠ ∅ there exists g ∈ G such that

d_{r}(1, g) < 2^{−n}
g · y′ ∈ U.

We may assume that d(U_{n}) < 2^{−n} for d the metric on X.

With this in hand we now wish to find a sequence of y_{0} = y, y_{1}, y_{2}, ... ∈ [y]_{G} and g_{0}, g_{1}, g_{2}, ... ∈ G such that:

(i) g_{i} · y_{i} = y_{i+1};

(ii) y_{i+1} ∈ U_{i};
(iii) d_{r}(g_{i+1}, 1) < 2^{−i}.

If this can be fulfilled then we are done, since for h_{i} = g_{i}g_{i−1}...g_{0} we have (h_{i})_{i∈ℕ} is d_{r}-Cauchy by right invariance and (iii), and so has a limit h_{∞}. Yet by (ii) we have

h_{i} · y = y_{i+1} → h_{∞} · y = x.

So suppose we have found the y_{j} and g_{j} for j ≤ i. Then since y_{i} ∈ U_{i} and U_{i} ∩ U_{i+1} ∩ [x]_{G} ≠ ∅ we can use assumption on the U_{i} to find some g_{i+1} ∈ G with

g_{i+1} · y_{i} ∈ U_{i+1}
d_{r}(1, g_{i+1}) < 2^{−i}.

Corollary 7.53. Every solvable Polish group has the weak Glimm-Effros property.

CHAPTER 8 The dark side

Here we take time with the non-Polishable groups and other pathologies. 8.2 gives a proof of the Kechris-Louveau theorem that there is a Borel equivalence relation not Borel reducible to the orbit equivalence relation of a continuous Polish group action. We also recall that there are non-Polishable Borel groups, that there are Borel groups which cannot be realized as Borel subgroups of Polish groups, and that there are even natural examples of abstract groups of cardinality the continuum with no Polish topology compatible with their group structure.

Definition 8.1. For x⃗ = (x_{0}, x_{1}, x_{2}, ...), y⃗ = (y_{0}, ...) ∈ ℝ^{ℕ} set x⃗E_{1}y⃗ if and only if there is some N ∈ ℕ so that

∀m > N(x_{m} = y_{m}).

Theorem 8.2. (Kechris, Louveau) If G is a Polish group and X is a Polish G-space then

E_{1} ≰_{B} E_{G}^{X}.

Proof. Suppose

θ : ℝ^{ℕ} → X

is Borel such that

∀x⃗, y⃗ ∈ ℝ^{ℕ}(x⃗E_{1}y⃗ ⇒ θ(x⃗)E_{G}^{X}θ(y⃗)).

For each n let H_{n} be ℝ^{n} in the usual topological and group structure. At each n we have ℝ^{ℕ} as a Polish H_{n}-space in the natural sense: for x⃗ ∈ ℝ^{ℕ} and h ∈ H_{n} define

(h · x⃗)(l)

to be h(l) + x_{l} for l < n and just x_{l} for l ≥ n. Moreover

E_{1} = ⋃_{n∈ℕ} E_{H_{n}}^{ℝ^{ℕ}}.

Fix d_{G} a complete metric on G.

Following 2.49 and 3.17 we can find a comeager C ⊂ ℝ^{ℕ} so:

(i) θ|_{C} : C → X is continuous;

(ii) for all W an open neighborhood of G, n ∈ ℕ, and x⃗ ∈ C, there is some open neighborhood V of the identity in H_{n} such that

∀^{∗}h ∈ V (∃g ∈ W (θ(h · x⃗) = g · θ(x⃗)));

(iii) and for notational convenience assume that there are open dense O_{i} ⊂ ℝ^{ℕ} such that

C = ⋂_{i∈ℕ} O_{i}.

For each n the set

C_{n} =_{df} {x⃗ ∈ ℝ^{ℕ} : ∀^{∗}h ∈ H_{n}(h · x ∈ C))}

is comeager by 2.47. There is a detail here: if

y⃗ ∈ ⋂_{m≥n} C_{m}

then

∀h ∈ H_{n}(h · y⃗ ∈ ⋂_{m≥n} C_{m}).

We may now find sequences of x⃗ = x⃗_{0}, x⃗_{1}, ... in ℝ^{ℕ}, open U_{0}, U_{1}, ... in ℝ^{ℕ}, and h_{0}, h_{1}, ... with each h_{n} ∈ H_{n}, and g_{0}, g_{1}, ... in G so that at each i

(iv) x⃗_{i+1} = h_{i} · x⃗_{i} ∈ U_{i+1} ∩ C ∩ ⋂{C_{j} : j > i};

(v) h_{i}(l) > 0 for each l < i;
(vi) h_{i}(l) < 2^{−i} for each l < i;
(vii) \overline{U}_{i+1} ⊂ U_{i} ⊂ O_{i};
(viii) g_{i} · θ(x) = θ(x_{i});
(ix) d_{G}(g_{i+1}, g_{i}) < 2^{−i}.

Once we do all this we are quickly done. By (ix) there exists some g_{∞} with

g_{i} → g_{∞}

and from (vi) there will be some x⃗_{∞} ∈ ℝ^{ℕ} with

x⃗_{i} → x⃗_{∞}.

By (v), x⃗_{∞} is not E_{1}-equivalent to x⃗_{0} = x⃗, whilst (vii) gives

x⃗_{∞} ∈ C

and so by (i)

θ(x⃗_{i}) = g_{i} · (θ(x⃗)) → θ(x⃗_{∞}) = g_{∞} · (θ(x⃗)),

so

θ(x⃗)E_{G}θ(x⃗_{∞}).

Now to prove such sequences exists. Suppose we have g_{j}, U_{j}, x⃗_{j} for j ≤ i and h_{j} for j < i. Start by fixing an open neighborhood W of 1_{G} so that for all g ∈ W

d_{G}(g_{i}, gg_{i}) < 2^{−i}.

Then by assumption of x⃗_{i} ∈ C we may find open V containing the identity of 1_{H_{i}} so that

∀^{∗}h ∈ V (∃g ∈ W (g · θ(x⃗_{i}) = θ(h · x⃗_{i}))).

Now we can find V′ ⊂ V open and non-empty so that for all h ∈ V′ and l ≤ i

h(l) > 0
h(l) < 2^{−i−1}
h · x⃗_{i} ∈ U_{i}.

By assumption

x⃗_{i} ∈ ⋂_{j≥i} C_{j}

we have that

∀^{∗}h ∈ V′(h · x⃗_{i} ∈ C).

Choosing h_{i} ∈ V′ meeting all the relevant comeager sets

x⃗_{i+1} = h_{i} · x⃗_{i} ∈ C

and there exists g ∈ W with

g · θ(x⃗_{i}) = θ(x⃗_{i+1}).

Taking g_{i+1} = gg_{i} and U_{i+1} a sufficiently small open set around x⃗_{i+1} continues the construction.

In particular this gives another proof of:

Theorem 8.3. (folklore) There are non-Polishable Borel groups.

Proof. Let c_{00} be the subgroup of ℝ^{ℕ} consisting of sequences with finite support, so that x⃗ ∈ c_{00} if there is some N so that

∀n > N(x_{n} = 0).

If τ was a Polish topology on this group, then the inclusion

(c_{00}, τ) ↪ ℝ^{ℕ}

would be continuous by 2.56, and so ℝ^{ℕ} would be a Polish c_{00}-space with orbit equivalence relation E_{1}, violating 8.2.

Theorem 8.4. There is a Borel group that cannot be realized as a Borel subgroup of a Polish group.

Proof. We begin with

A =_{df} {(y, x) ∈ 2^{ℕ} × 2^{ℕ×ℕ} : ∃n(y = x(n, ·))}.

Let τ_{0} and τ_{1} be the usual product topologies on 2^{ℕ} and 2^{ℕ×ℕ} respectively.

Claim(I): There are no Polish topologies τ̂_{0} and τ̂_{1} on 2^{ℕ} and 2^{ℕ×ℕ} compatible with the usual Borel structure in which A becomes closed in the product.

Proof of claim. Otherwise let τ̂_{0} and τ̂_{1} be the topologies and consider the Borel injections:

i_{0} : (2^{ℕ}, τ_{0}) → (2^{ℕ}, τ̂_{0}),
i_{1} : (2^{ℕ×ℕ}, τ_{1}) → (2^{ℕ×ℕ}, τ̂_{1}),

and go to comeager C_{0} ⊂ 2^{ℕ} and C_{1} ⊂ 2^{ℕ×ℕ} on which these maps are continuous. Viewing 2^{ℕ×ℕ} as an S_{∞}-space as at 2.22, where (g · x)(n, m) = x(g^{−1}(n), m), we can use Kuratowski-Ulam to find some x ∈ C_{1} so that

∀^{∗}g ∈ S_{∞}(g · x ∈ C_{1})

and [x]_{S_{∞}} is dense in 2^{ℕ×ℕ}. But then for y = x(0, ·) we have

{y} × ([x]_{S_{∞}}) ∩ C_{1} ⊂ A

is dense in {y}×C_{1}. Since A is relatively closed this would assert that for all x̂ ∈ C_{1}

∃n((x̂(n, ·) = x(0, ·)),

which flies in the face of

{x̂ ∈ 2^{ℕ×ℕ} : (x̂(n, ·) = x(0, ·)}

being meager for each n. (Claim(I)□)

Define F (2^{ℕ}) and F (2^{ℕ×ℕ}) to be the free groups on the generating sets 2^{ℕ} and 2^{ℕ×ℕ} with the natural Borel structure. These are standard Borel spaces, since we may identify them with a Borel subsets of

(2^{ℕ} ∪ (2^{ℕ})^{−1})^{<ℕ} ∪ {1}

and

(2^{ℕ×ℕ} ∪ (2^{ℕ×ℕ})^{−1})^{<ℕ} ∪ {1},

where (·)^{−1} is understood as providing a formal inverse and (2^{ℕ})^{−1} and (2^{ℕ×ℕ})^{−1} are viewed as homeomorphic copies of 2^{ℕ} and 2^{ℕ×ℕ}. For y ∈ 2^{ℕ} and x ∈ 2^{ℕ×ℕ} let

ψ_{y}(x) = x

if (y, x) ∈ A and

ψ_{y}(x) = x^{−1}

otherwise. By the defining property of the free group this extends to homomorphism

ψ_{y} : F (2^{ℕ×ℕ}) → F (2^{ℕ×ℕ}).

Thus we have a map

ψ : 2^{ℕ} → Hom(F (2^{ℕ×ℕ}))
y ↦ ψ_{y}.

Again by the property of being a free group this extends to a homomorphism

ψ : F (2^{ℕ}) → Hom(F (2^{ℕ×ℕ})).

Let H = F (2^{ℕ×ℕ}) ⋊_{ψ} F (2^{ℕ}) be the semi-direct product along ψ. So H will have underlying set F (2^{ℕ×ℕ}) × F (2^{ℕ}) and group operation

(g_{0}, h_{0})(g_{1}, h_{1}) = (g_{0}ψ_{h_{0}}(g_{1}), h_{0}h_{1}).

Since the set A is Borel the map

(g, h) → ψ_{h}(g)

is Borel and so H is a Borel group. It suffices then to see:

Claim(II) H is not a Borel subgroup of any Polish group.

Proof of Claim: Suppose otherwise and let (G, τ) be such a Polish group. By continuity of multiplication the set

{(g, ĝ) ∈ G × G : gĝg^{−1}ĝ^{−1} = 1}

is τ-closed. Then by 2.3 we can find τ̂ ⊃ τ a richer Polish topology in which

B_{0} =_{df} {1} × 2^{ℕ}
B_{1} =_{df} 2^{ℕ×ℕ} × {1}

are both closed.

So let τ̂_{0} and τ̂_{1} be the Polish topologies on 2^{ℕ} and 2^{ℕ×ℕ} under their natural identification with the (closed) subspaces B_{0} and B_{1} of (G, τ̂). But then A in the product topology for (2^{ℕ}, τ̂_{0}) and (2^{ℕ×ℕ}, τ̂_{1}) is closed since it can be identified with the set

{(g, h) ∈ B_{0} × B_{1} : gĝg^{−1}ĝ^{−1} = 1}

contradicting claim(I). (Claim(II)□)

Exercise 8.5. (See 6.2 of [4].) Every Σ ∼_{1}^{1} equivalence relation is induced by a Borel action of a Borel group.

Exercise 8.6. Let B_{∞} be the Baire category analogue of M_{∞}. More precisely, let B_{∞} be the class of Borel bijections on 2^{ℕ} which are homeomorphisms on some comeager subset of 2^{ℕ}, with identification of π_{1} and π_{2} whenever

∀^{∗}x ∈ 2^{ℕ}(π_{1}(x) = π_{2}(x)).

The next few steps indicate that there is no Polish group topology on B_{∞} compatible with its algebraic structure: Suppose for a contradiction that (B_{∞}, τ) is a Polish group. Let B(B_{∞}, τ) be the Borel structure on B_{∞} generated by this Polish topology and Σ ∼_{1}^{1}(B_{∞}, τ) the resulting collection of Σ ∼_{1}^{1} sets.

(i) For U ⊂ 2^{ℕ} open let

A(U) = {π ∈ B_{∞} : π|_{U} = id|_{U}}.

Each such A(U) ∈ B(B_{∞}, τ). (For suitably chosen \{π_{i} : i ∈ ℕ\} ∈ A(X \ \overline{U}) we have

π ∈ A(U) ⇔ ∀i(π_{i} ∘ π = π ∘ π_{i}).)
(ii) For U, V ⊂ 2^{ℕ} open let
B(U, V) = {π ∈ B_{∞} : π[U] ⊂ V }.

If U and V are open then the set B(U, V) is Σ ∼_{1}^{1}(B_{∞}, τ). (By translating V it suffices to show that B(U, U) ∈ Σ ∼_{1}^{1}(B_{∞}, τ). But if (W_{n})_{n∈ℕ} is a basis for X, then π_{0} ∈ B(U, U) if and only if for all n

W_{n} \ \overline{U} = ∅

or

∃π_{1} ∈ A(X \ \overline{W_{n} ∪ U})(π_{1} ∘ π_{0} ∈ A(U)).)

(iii) For {V_{n} : n ∈ ℕ} a basis of 2^{ℕ} and U ≠ ∅, 2^{ℕ} open there will be some V_{n} ≠ ∅, 2^{ℕ} with B(U, X \ \overline{V_{n}}) non-meager.

(iv) Then (iii) implies for any open U, V ≠ ∅, 2^{ℕ} that B(U, V) is non-meager.

(v) Let

C(U, V) = {π ∈ B_{∞} : π[U] ⊂ V, ≠ V }

and

D(U, V) = {π ∈ B_{∞} : π[U] = V }

for open U, V ≠ ∅, 2^{ℕ}. (ii) implies that all these sets have Baire property. Use the separability of τ to conclude that for no such U and V is D(U, V) open.

(vi) Conclude that for each open U ≠ ∅, 2^{ℕ} the set C(U, U) is non-meager. (vii) Now note that C(U, U) is open for U as above. (First from 2.56(iv) we get

that C(U, U) contains some non-empty open set, and then that any two elements of C(U, U) may be transformed one to the other by left and right multiplication from D(U, U).)

(viii) For U as above and W_{0} and W_{1} open, disjoint, and non-empty, and not meeting U, and for π_{0}, π_{1} ∈ B_{∞} with

π_{i}[U] = U ∪ W_{i},

we have that

π_{i}[C(U, U)] =_{df} {π_{i} ∘ π : π ∈ C(U, U)} = C(U, U ∪ W_{i})

are open for i = 0, 1, and

∴ D(U, U) = C(U, U ∪ W_{0}) ∩ C(U, U ∪ W_{1})

is open, to the grief of (v).

154 8. THE DARK SIDE

CHAPTER 9 Beyond Borel

In various special cases chapter 6 proved the equivalence of classifiable by countable structures and not reducing a turbulent orbit equivalence relation. Many of the technical difficulties arising there as a result of the desire to work solely in the Borel category vanish when considering reductions that are Δ ∼_{2}^{1} or even only in L(ℝ); for instance, the process that extracts the code for a Scott sentence from a countable model is Δ ∼_{2}^{1}, as well as universally Baire measurable in the codes, but not Borel. Allowing the most general class of reductions for which we still have regularity properties such as Baire measurability perhaps matches more closely the vague concept of effective or definable cardinality. In measuring the effective cardinality or classification difficulty of a collection of objects we should presumably be willing to use any reasonably well behaved class of definable functions; we would probably be relatively unconcerned with the minutiae of whether a specific reduction can be arrived at in a certain complexity class.

The present chapter accordingly considers the theory of reductions much more complicated than Borel, but still having a plausible claim on being reasonable or definable.

Below §9.1 proves general results for both the generic ergodicity and turbulence. The first of these has similarities with recent work on Ulm-type classifications.

§9.2 considers cardinalities in the universe L(ℝ) of all objects generated from the reals by transfinite iterations of definable subsets. These results would seem to be devoid of “practical application”, and are included for the philosophical interest. The preference for countable models is given support by a proof that if we can assign any kind of discrete well orderable structures as a complete invariant to E_{G}^{X} then we can do so with countable structures.

This chapter will be unreadable without a substantial background in modern set theory.

9.1. Two theorems by transfinite changes in topologies

Definition 9.1. A function

θ : ℕ^{ℕ} → HC

is absolutely Δ ∼_{2}^{1} if there is some parameter z ∈ ℕ^{ℕ} and formula ψ such that: (I) for all x ∈ ℕ^{ℕ} and y ∈ HC

θ(x) = y

if and only if there is some countable transitive set M including x, y, and z such that

(M, ∈) ⊧ ψ(x, y, z);

(II) this absolutely defines a function, in the sense that in all generic extensions it continues to be the case that for all x ∈ ℕ^{ℕ} there exists y ∈ HC and countable transitive M with

(M, ∈) ⊧ ψ(x, y, z).

We can then extend this to functions between arbitrary Polish spaces in the natural manner. For X_{1} and X_{2} Polish spaces,

σ_{1} : X_{1} → ℕ^{ℕ}
σ_{2} : X_{2} → ℕ^{ℕ}

Borel bijections, we may say that θ : X_{1} → X_{2} is absolutely Δ ∼_{2}^{1} if the induced

σ_{2} ∘ θ ∘ (σ_{1})^{−1} : ℕ^{ℕ} → ℕ^{ℕ}

is absolutely Δ ∼_{2}^{1} in the sense above. This further definition does not depend on the choice of σ_{1} and σ_{2}.

Similarly for X Polish and

σ : X → ℕ^{ℕ}

a Borel bijection, we may say that θ : X → HC is absolutely Δ ∼_{2}^{1} if the induced

θ ∘ (σ)^{−1} : ℕ^{ℕ} → HC

is absolutely Δ ∼_{2}^{1}.

This definition adapts for functions whose domain is a natural subclass of the hereditarily countable sets. For instance a function

π : ω_{1} → HC

is absolutely Δ ∼_{2}^{1} if there is some parameter z ∈ ℕ^{ℕ} and formula ψ such that: (I) for all countable ordinals α and y ∈ HC

θ(α) = y

if and only if there is some countable transitive model M including α, y, and z such that

M ⊧ ψ(α, y, z);

(II) through all generic extensions it continues to be the case that for all countable α there exists y ∈ HC and countable transitive M with

M ⊧ ψ(α, y, z).

For equivalence relations E and F

E ≤_{aΔ_{2}^{1}} F

indicates there is some absolutely Δ ∼_{2}^{1} function θ so that for all x_{1}, x_{2}

x_{1}Ex_{2} ⇔ θ(x_{1})F θ(x_{2}).

Similarly E ≤_{aΔ_{2}^{1}} id(2^{<ω_{1}}) indicates that there is an absolutely Δ ∼_{2}^{1} function assigning bounded subsets of ω_{1} as complete invariants and E ≤_{aΔ_{2}^{1}} id(HC) states that we may in an absolutely Δ ∼_{2}^{1} fashion assign hereditarily countable sets as complete invariants.

Theorem 9.2. (folklore) Any absolutely Δ ∼_{2}^{1} function is Baire measurable.

Proof. By the devices of [78].

We need to show that the absolutely Δ ∼_{2}^{1} function is continuous on a comeager set.

It suffices to prove this for functions between uncountable recursive Polish spaces, so let z and ψ be the witnesses to

θ : 2^{ℕ} → 2^{ℕ}

absolutely Δ ∼_{2}^{1} and choose some countable transitive

X≺V_{ω+ω}

containing z. Form N as the transitive collapse of X. There will be a comeager set of C ⊂ 2^{ℕ} of reals Cohen generic over N. For any x ∈ C

N[x] ⊧ ∃y, M(M ⊧ ψ(x, y, z))

and hence

θ(x) = y

since M must be well founded in V.

Now, as follows from the usual development of forcing, for each basic open U ⊂ 2^{ℕ} there will be some open V ⊂ 2^{ℕ} so that for all x ∈ C

N[x] ⊧ ∃y ∈ U∃M(M ⊧ ψ(x, y, z))

if and only if

x ∈ V,

for we can simply let V be the union of the conditions forcing that θ(x) is in U.

Since absolutely Δ ∼_{2}^{1} includes Borel and is closed under composition, every absolutely Δ ∼_{2}^{1} function is universally Baire measurable.

Theorem 9.3. (folklore) If

π : ω_{1} → 2^{ℕ}

is absolutely Δ ∼_{2}^{1}, then there are α < β < ω_{1} with

π(α) = π(β).

Proof. (Compare similar arguments in [20].) Fix the parameter z and the formula ψ witnessing π absolutely Δ ∼_{2}^{1}. Thus z and ψ will continue to define a function through all generic extensions, and by Shoenfield absoluteness it suffices in some extension to find α ≠ β with

π(α) = π(β).

Let ℙ =Coll(ω, (2^{ℵ_{0}})^{+}). For α < (2^{ℵ_{0}})^{+} let τ_{α} be the term for π(α) as calculated in V^{ℙ}, in the sense that

ℙ ⊩ ∃ transitive M, M ⊧ ψ(α, τ_{α}[Ġ], z).
Claim: For all n ∈ ℕ, p_{0}, p_{1} ∈ ℙ,
p_{0} ⊩_{ℙ} (τ_{α}[Ġ])(n) = 0

excludes

p_{1} ⊩_{ℙ} (τ_{α}[Ġ])(n) = 1.

Proof of claim. Otherwise let G_{l}, G_{r} ⊂ ℙ be mutually generic filters below p_{0}, p_{1} respectively. Then by the definability assumption on π

V [G_{l}] ⊧ ∃ transitive M, M ⊧ ψ(α, τ_{α}[G_{l}], z),

then this goes up to its generic extension so that

V [G_{l}, G_{r}] ⊧ ∃ transitive M, M ⊧ ψ(α, τ_{α}[G_{l}], z),

and so

V [G_{l}, G_{r}] ⊧ π(α) = τ_{α}[G_{l}].

Similarly on the V [G_{r}] side we obtain

V [G_{l}, G_{r}] ⊧ π(α) = τ_{α}[G_{r}],

and a contradiction to

(τ_{α}[G_{l}])(n) = 0 ≠ 1 = (τ_{α}[G_{r}])(n).
(Claim□)

Thus we obtain for any generic G ⊂ ℙ that

(τ_{α}[G])(n) = 0

if and only if there is some p_{0} ∈ ℙ with

p_{0} ⊩_{ℙ} (τ_{α}[Ġ])(n) = 0.

In particular, this element of Cantor space is already in V. Thus by the pigeonhole principle we may find α ≠ β < (2^{ℵ_{0}})^{+} with τ_{α}[G] = τ_{β}[G].

A nearly identical proof leads to the following result:

Theorem 9.4. (folklore) If

π : ω_{1} → P_{ℵ_{0}}(2^{ℕ})

is absolutely Δ ∼_{2}^{1}, then there are α < β < ω_{1} with

π(α) = π(β).

Proof. Essentially 9.3 used the claim established in the course of its proof to see that if τ is a ℙ term for a real, and in the product forcing we have

ℙ × ℙ ⊩ τ[Ġ_{l}] × τ[Ġ_{r}],

where Ġ_{l} × Ġ_{r} names the generic object on the product forcing notion, then there is some a in the ground model with

ℙ ⊩ a = τ[Ġ].

However one can easily prove by transfinite induction on the rank of the terms involved that whenever

(p_{1}, p_{2}) ⊩_{ℙ×ℙ} τ_{1}[Ġ_{l}] × τ_{2}[Ġ_{r}]

then there will be some a ∈ V and p_{3} ≤ p_{1} with p_{3} ⊩ a = τ[Ġ]. The point is that if

(p_{1}, p_{2}) ⊩_{ℙ×ℙ} τ_{1}[Ġ_{l}] × τ_{2}[Ġ_{r}],

and σ_{1} is a term for an element of τ_{1}, then there will be some σ_{2}, a term for an element of τ_{2}, and q_{1} ≤ p_{1}, q_{2} ≤ p_{2} with

(q_{1}, q_{2}) ⊩_{ℙ×ℙ} σ_{1}[Ġ_{l}] × σ_{2}[Ġ_{r}],

and since we may assume that the terms σ_{1} and σ_{2} have lower rank than τ_{1} and τ_{2} we can apply the inductive hypothesis.

Theorem 9.5. Let G be a Polish group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{aΔ_{2}^{1}} id(2^{<ω_{1}});

2. there is a properly generically ergodic Polish G-space Y and a continuous G-embedding π : Y → X.

Proof. The incompatibility of 1. and 2. is a known result, and in some form can be found in [20] or [37]. From the conjunction of the two and 9.2 we obtain a comeager set C ⊂ Y and a continuous in the codes reduction on C to id(2^{<ω_{1}}). Using the boundedness principle for Σ ∼_{1}^{1}, there will be a single δ < ω_{1} for which the reduction on C provides subsets of γ as complete invariants. Since γ is countable this gives a Baire measurable reduction of the orbit equivalence relation E_{G}^{Y} to id(ℝ), contradicting 3.3.

Instead we concentrate on showing at least one to be true. The technique is to define a transfinite sequence of topologies around any orbit in the space. The proof resembles the method of §6.2. The process will be canonical and in some sense absolutely Δ ∼_{2}^{1}. Using 9.3 it will be seen that for any point the transfinite process must terminate. We will then obtain a dichotomy based on whether the topology terminates with the subjected orbit a dense G_{δ}. If for every orbit it is so, then we will be able to produce a bounded subset of ω_{1} as a code; if for some [x]_{G} not we obtain the set of points with the same closure as [x]_{G} forming a properly generically ergodic Polish G-space.

As in 7.32 fix a basis B = {U_{l} : l ∈ ℕ} of X, {g_{i} : i ∈ ℕ} = G_{0} a countable dense subgroup of G, B_{0} = {V̂_{k} : k ∈ ℕ} a countable basis for G including all sets of the form {g ∈ G : d(g, g_{i}) < q} for i ∈ ℕ and q ∈ ℚ, and {V_{n} : n ∈ ℕ} a neighborhood basis at 1_{G} with V_{n} = (V_{n})^{−1} all n.

Focusing on some particular x ∈ X we assign to each countable ordinal α a basis B_{α}^{x} of a Polish topology τ_{α}^{x} on X. We will also produce Polish spaces (X_{α}^{x}, τ_{α}^{x}) in which every orbit is dense.

From τ_{α+1}^{x} we let X_{α+1}^{x} be the set of y ∈ X_{α}^{x} whose orbit has the same τ_{α+1}^{x} closure as [x]_{G}. At λ at limit we take

X_{λ}^{x} = ⋂_{α<λ} X_{α}^{x}.

So in particular for α_{1} < α_{2} the space X_{α_{1}}^{x} includes X_{α_{2}}^{x}.

We will maintain inductively for α_{1} < α_{2} that

B_{α_{1}}^{x} ⊂ B_{α_{2}}^{x}

and at each α

(X, τ_{α}^{x}) is a Polish G − space

and mandate the following closure on B_{α}^{x}: for all W_{0}, W_{1} ∈ B_{0} and U, Û ∈ B_{α}^{x}

((X \ U)^{∗W_{0}} ∩ Û)^{ΔW_{1}} ∈ B_{α}^{x}.

We may assume that the basis B has these closure properties by 7.48 and so just let the process begin with

B_{0}^{x} = B.

At limit stages we take

B_{λ}^{x} = ⋃_{α<λ} B_{α}^{x}.

Note that at limit λ every y ∈ X_{λ}^{x} has orbit with the same τ_{λ}^{x}-closure as [x]_{G}.

All the work happens at successor steps and requires a split into cases. For that purpose at each n ∈ ℕ let O_{x,n,α} be the union of all open U ∈ B_{α}^{x} such that for all z ∈ X_{α}^{x} ∩ U and Û ∈ B_{α}^{x} with Û ∩ U ≠ ∅

∃g_{i} ∈ V_{n}(g_{i} · z ∈ Û ∩ U));

note there that O_{x,n,α} is defined by a Π ∼_{1}^{1} union of basic open sets in B_{α}^{x}.

Case(A): For some n the open set O_{x,n,α} is not dense in (X_{α}^{x}, τ_{α}^{x}).

Then let

B_{α+1}^{x} = B_{α}^{x};

from which it follows that for all γ > α

B_{γ}^{x} = B_{α}^{x}.

Claim(1): There is no non-meager orbit in (X_{α}^{x}, τ_{α}^{x}).

Proof of claim: Suppose instead that [y]_{G} is a non-meager orbit in (X_{α}^{x}, τ_{α}^{x}). By 7.12 [y]_{G} is G_{δ}. Following 7.50 applied to the G_{δ} orbit [y]_{G} we may find around each y′ ∈ [y]_{G} a basic open set in U_{y′} in (X_{α}^{x}, τ_{α}^{x}) so that for all z ∈ U_{y′} and Û ∈ B_{α}^{x} with

U_{y′} ∩ Û ≠ ∅

there will exist g ∈ V_{n} with

g · z ∈ U_{y′} ∩ Û.

Since [y]_{G} is dense in (X_{α}^{x}, τ_{α}^{x}) by the definition of the space, we have a contradiction to case assumption. (Claim(1)□)

So by 7.7 we are in alternative 2. of the theorem with Y = (X_{α}^{x}, τ_{α}^{x}).

Case(B): For each n the set O_{x,n,α} is dense in (X_{α}^{x}, τ_{α}^{x}) but [x]_{G} is not a G_{δ} set in (X_{α}^{x}, τ_{α}^{x}).

Claim(2) There is some n with

x ∈ (X \ O_{x,n,α})^{ΔG}.

Proof of claim: Since [x]_{G} is meager we may find a sequence (C_{k})_{k∈ℕ} of τ_{α}^{x} closed nowhere dense sets in (X_{α}^{x}, τ_{α}^{x}) so that

[x]_{G} ⊂ ⋃_{k∈ℕ} C_{k}.

Thus we may find a basic open V containing 1_{G} and x′ ∈ [x]_{G} such that for some k

∀g ∈ V (g · x′ ∈ C_{k}).

Choose n so that

(V_{n})^{2} ⊂ V

and then it follows from the nowhere density of C_{k} and the definition of O_{x,n,α} that for all g ∈ V_{n}

g · x′ ∈ X \ O_{x,n,α}.
(Claim(2)□)

Now take

B_{α+1}^{x} = B^{+}(B_{α}^{x}, B_{0}, {X \ O_{x,n,α} : n ∈ ℕ}).

The definition of the B^{+} operation will allow us to choose canonically from a wellordering of B_{α}^{x} a wellordering of B_{α+1}^{x}; each O_{x,n,α} can be thought of as being labeled by an ordinal α and a natural number n.

Case(C) For each n the set O_{x,n,α} is dense in (X_{α}^{x}, τ_{α}^{x}) and [x]_{G} is a G_{δ} set in (X_{α}^{x}, τ_{α}^{x}).

We then simply set B_{α+1}^{x} = B_{α}^{x}, from which for all γ > α

B_{γ}^{x} = B_{α}^{x}.

This completes the description of B_{α+1}^{x} from B_{α}^{x}.

Claim(3): For each x ∈ X there is some α < ω_{1} with

B_{α+1}^{x} = B_{α}^{x}.

Proof of claim: Suppose otherwise. Then at every α < ω_{1} we are in case (B) and [x]_{G} is meager in (X_{α}^{x}, τ_{α}^{x}).

From claim(2) there will be some basic open neighborhood of the identity, V_{m} ∈ B_{0}, and x_{0} ∈ [x]_{G} so that

x_{0} ∈ (X \ O_{x,n,α})^{∗V_{m}}.

Then choose m(0), m(1) so that

V_{m(1)}(V_{m(1)})^{−1} ⊂ V_{m(0)}
V_{m(0)}V_{m(1)} ⊂ V_{m}.

Subclaim(3a): x_{0} ∈ ((X \ O_{x,n,α})^{∗V_{m(0)}})^{∗V_{m(1)}} Proof of subclaim: Otherwise there is some g ∈ V_{m(1)} with

g · x_{0} ∈ (O_{x,n,α})^{ΔV_{m(0)}}
∴ x_{0} ∈ (O_{x,n,α})^{ΔV_{m(0)}g} ⊂ (O_{x,n,α})^{ΔV_{m}}.

(Subclaim(3a)□)

Thus in particular x_{0} ∈ ((X \ O_{x,n,α})^{∗V_{m(0)}})^{ΔV_{m(1)}}. Subclaim(3b): ((X \ O_{x,n,α})^{∗V_{m(0)}})^{ΔV_{m(1)}} ⊂ (X \ O_{x,n,α})^{∗V_{m(1)}} ⊂ X \ O_{x,n,α}. Proof of subclaim: The second inclusion follows by the continuity of the action

with respect to τ_{x}^{α}. For the first, note that if z ∈ (O_{x,n,α})^{ΔV_{m(1)}} then for all g ∈ V_{m(1)}

g · z ∈ (O_{x,n,α})^{ΔV_{m(1)}g^{−1}} ⊂ (O_{x,n,α})^{ΔV_{m(0)}}.

(Subclaim(3b)□)

Thus the set U_{x,n,m(0),m(1),α} of z ∈ (X_{α}^{x}, τ_{α}^{x}) for which

z ∈ ((X \ O_{x,n,α})^{∗V_{m(0)}})^{ΔV_{m(1)}}

becomes open in τ_{α+1}^{x} and has non-empty intersection with [x]_{G}; note that

U_{x,n,m(0),m(1),α} ∩ [x]_{G}

is not an open set in ([x]_{G}, τ_{α}^{x}) by subclaim 3(b) and density of the open O_{x,n,α}. Let n(x, α), m(0, x, α), m(1, x, α) be least such that

U_{x,n(x,α),m(0,x,α),m(1,x,α),α} ∩ [x]_{G}

not open in ([x]_{G}, τ_{α}^{x}). Then let

π̂_{x}(α) = {l ∈ ℕ : V̂_{l} · x ⊂ U_{x,n(x,α),m(0,x,α),m(1,x,α),α}}.

The set of g ∈ G with

g · x ∈ U_{x,n(x,α),m(0,x,α),m(1,x,α),α}

is open by (X_{α}^{x}, τ_{α+1}^{x}) being a Polish G-space. The π̂_{x} is absolutely Δ ∼_{2}^{1} since π̂_{x}(α) is uniformly definable from α over the α + 1th admissible structure containing x and an appropriate code for G, X, and the action of G on X.

Subclaim(3c): For β < α we have π̂_{x}(α) ≠ π̂_{x}(β).

Proof of subclaim: As remarked above, U_{x,n,m(0),m(1),γ} ∩ [x]_{G} is τ_{δ}^{x}-open if and only if γ < δ. This gives that

U_{x,n,m(0),m(1),α} ∩ [x]_{G} = ⋃_{l∈π̂_{x}(α)}(V̂_{l} · x),

U_{x,n,m(0),m(1),β} ∩ [x]_{G} = ⋃_{l∈π̂_{x}(β)}(V̂_{l} · x),

and

U_{x,n,m(0),m(1),α} ∩ [x]_{G} ≠ U_{x,n,m(0),m(1),β} ∩ [x]_{G}.

Thus π̂_{x}(α) ≠ π̂_{x}(β). (Subclaim(3c)□) This then gives a contradiction to 9.3. (Claim(3)□) So it terminates. If the process ever finishes in case(A) then we are done

and have 2. Instead suppose it to unfailingly finish in (C). We will produce the absolutely Δ ∼_{2}^{1} reduction. I will only concentrate on formally defining

θ : X → 2^{<ω_{1}}

and not verifying that it has the correct level of complexity. This last point is similar to say the proof of L∩HC having a Δ_{2}^{1} in the codes wellordering.

Fix x ∈ X and let α(x) be the ordinal at which for first time B_{α(x)}^{x} = B_{α(x)+1}^{x}; by case assumption [x]_{G} is a dense G_{δ} in (X_{α(x)}^{x}, τ_{α(x)}^{x}). For each α ≤ α(x) we describe a countable set A(x, α) of countable ordinals, an ordinal γ(x, α) =_{df} supA(x, α), and an onto function

F_{α}^{x} : A(x, α) ↠ B_{α}^{x}.

The function only depends on the orbit equivalence class of x and for α < β

F_{α}^{x} ⊂ F_{β}^{x}.

At limit stages take unions and at α = 0 just begin with γ(x, 0) = ω and a function provided by some countable enumeration of B. All the content in the definition takes place at successor steps.

So suppose that F_{α}^{x} : A(x, α) → B_{α}^{x} has been defined and we wish to continue to F_{α+1}^{x}. We define

F_{α}^{x} = F_{α,0}^{x} ⊂ F_{α,1}^{x} ⊂ F_{α,2}^{x} ⊂ ...

with A(x, α, i) the the domain of F_{α,i}^{x}, γ(x, α, i) the supremum of A(x, α, i), and take

F_{α+1}^{x} = ⋃_{i∈ℕ} F_{α,i}^{x}

A(x, α + 1) = ⋃_{i∈ℕ} A(x, α, i).

Fix some Gödel pairing function

s : ω_{1}^{<ω} → ω_{1}

which is injective and uniformly Δ_{1} inside any admissible set. Given any n ∈ ℕ let

F_{α,1}^{x}(γ(x, α, 0) + s(⟨n⟩)) = O_{x,n,α}.

We continue this at the later stages, making room for the Δ Vaught transforms, intersections and unions. If U, U′ ∈ B_{α+1}^{x} with

F_{α,i}^{x}(δ) = U
F_{α,i}^{x}(δ′) = U′

let

F_{α,i+1}^{x}(γ(x, α, i) + s(⟨0, δ, δ′l(0), ..., l(k), l⟩)) =
(U ∩ (X \ U′)^{∗W_{l(0)}} ∩ ... ∩ (X \ U′)^{∗W_{k(l)}})^{ΔW_{l}}
F_{α,i+1}^{x}(γ(x, α, i) + s(⟨1, δ, δ′⟩)) = U ∩ U′

and

F_{α,i+1}^{x}(γ(x, α, i) + s(⟨2, δ, δ′⟩)) = U ∪ U′.

We take A(x, α, i + 1) to be the union of A(x, α, i) and the new ordinals defined to be in the domain of F_{α,i+1}^{x} by the above.

Finally we need to describe

θ(x) ∈ 2^{<ω_{1}}.

But we can let θ(x) be the set of s(⟨α, δ⟩) such that α ≤ α(x) and δ ∈ A(x, α) and for

U = F_{α}^{x}(δ)

in the space (X_{α}^{x}, τ_{α}^{x}) we have

U ∩ [x]_{G} ≠ ∅.

From the definition of these objects the assignment

x ↦ θ(x)

is an invariant of the orbit. We do need to worry about some other E_{G}^{X}-inequivalent y ∈ X receiving the same invariant, but any such [y]_{G} cannot also be a dense G_{δ} in

(X_{α(x)}^{x}, τ_{α(x)}^{x});

whence it is either not dense in some

(X_{α}^{x}, τ_{α}^{x})

for α ≤ α(x) or else

α(x) < α(y).

In either case

θ(x) ≠ θ(y).

Corollary 9.6. (Becker; Hjorth, Kechris) Let G be a Polish group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{aΔ_{2}^{1}} id(2^{<ω_{1}});

2. E_{0} ⊑_{c} E_{G}^{X}.

The original proof of 9.6 can be found in [37], bears no resemblance to 9.5, and fails to give the further information obtained above. On the other hand this earlier argument showed that in 9.6(1.) we can draw the stronger conclusion that there is a C-measurable reduction to id(2^{<ω_{1}}). [37] also mentions that we may not in general replace 1. above by E_{G}^{X} ≤_{aΔ_{2}^{1}} id(ℝ) or even E_{G}^{X} ≤_{L(ℝ)} id(ℝ).

Theorem 9.7. Let G be a Polish group and X a Polish G-space. Then exactly one of the following holds:

1. E_{G}^{X} ≤_{aΔ_{2}^{1}} id(HC);

2. there is a turbulent Polish G-space Y and a continuous G-embedding π : Y → X.

Proof. This is near to the proof of 9.5 so we should concentrate on the differences. At most one of the alternatives above can hold by 3.18.

For establishing at least one, we again define a transfinite sequences of topologies τ_{x}^{α} with bases B_{α}^{x} so that for α_{1} < α_{2}

B_{α_{1}}^{x} ⊂ B_{α_{2}}^{x}.

Again it will stop the moment [x]_{G} becomes G_{δ} in τ_{α}^{x}, and again (I) corresponds to every x thus ending as a G_{δ}. Again at limits

τ_{λ}^{x} = ⋃_{α<λ} τ_{α}^{x},

X_{λ}^{x} = ⋂_{α<λ} X_{α}^{x}.

Again we will maintain inductively that if W_{0}, W_{1} ⊂ G are basic open sets with U, U′ ∈ B_{α}^{x} then

((X \ U)^{∗W_{0}} ∩ U′)^{ΔW_{1}} ∈ B_{α}^{x}.

The change is in the manufacture of τ_{α+1}^{x} from τ_{α}^{x}. It now becomes a multi step process, with an intermediate stage given by the generalized Scott analysis.

From τ_{α}^{x} we produce τ_{α,0}^{x} as an intermediate topology according to 6.31. Letting φ_{x}^{τ_{α}^{x}} be the generalized Scott sentence calculated in (X_{α}^{x}, τ_{α}^{x}) for x, we define the intermediate space

X_{α,0}^{x} =_{df} {y ∈ X_{α}^{x} : φ_{x}^{τ_{α}^{x}} = φ_{y}^{τ_{α}^{x}}}.

We then take

τ_{α,0}^{x} =_{df} τ(φ_{x}^{τ_{α}^{x}})

to be the topology with basis

B_{α,0}^{x} = B_{α}^{x} ∪ {U(φ_{γ(x)}^{τ_{α}^{x}}(x̂, U, V_{n})) : n ∈ ℕ, U ∈ B_{α}^{x}, x̂ ∈ [x]_{G} ∩ U} ⊃ B_{α}^{x},

where γ(x) is defined as at 6.23 in the space (X_{α}^{x}, τ_{α}^{x}), using the basis B_{α}^{x}, and

U(φ_{β}^{τ_{α}^{x}}(x̂, U, V_{n})) = {z ∈ X_{α}^{x} ∩ U : φ_{β}^{τ_{α}^{x}}(x̂, U, V_{n}) = φ_{β}^{τ_{α}^{x}}(z, U, V_{n})}

for any β < ω_{1}.

Case(A): There is no G_{δ} orbit in (X_{α,0}^{x}, τ_{α,0}^{x}).

Thus by 6.37 and 6.35 we have that the action of G on (X_{α,0}^{x}, τ_{α,0}^{x}) is turbulent. Since the topology τ_{α,0}^{x} is richer than the original topology we have that some turbulent orbit equivalence relation allows a continuous G-embedding into X and we finish.

Case(B): There is a G_{δ} orbit in (X_{α,0}^{x}, τ_{α,0}^{x}).
Case(B1): [x]_{G} is that G_{δ} orbit.
Then we at once stop the process. We let B_{α+1}^{x} = B_{α,0}^{x}, and for all γ > α
B_{γ}^{x} = B_{α+1}^{x}.
Case(B2): [x]_{G} is not a G_{δ} orbit.

Let y ∈ X_{α,0}^{x} be the point with a G_{δ} orbit. For each n ∈ ℕ and y′ ∈ [y]_{G} using 7.12 as 7.49 we may choose some basic open (in the τ_{α,0}^{x} topology) set U_{y′,n} ∈ B_{α,0}^{x} containing y′ such that for all y_{0} ∈ [y]_{G} and open U with U ∩ U_{y′,n} non-empty (in X_{α,0}^{x}) we have

∃g ∈ V_{n}(g · y_{0} ∈ U).

Claim(1): For U_{y′,n} as above and U open with

U ∩ U_{y′,n} ≠ ∅,

and for W_{0}, W_{1} ⊂ G in B_{0} with

V_{n}W_{1}W_{1}^{−1} ⊂ W_{0}

we have that the set

((X_{α,0}^{x} \ U)^{∗W_{0}} ∩ U_{y′,n})^{ΔW_{1}}

is not relatively open in [x]_{G}.

Proof of claim: Suppose instead U′ ∈ B_{α,0}^{x} has non-empty intersection with [x]_{G} and

[x]_{G} ∩ U′ ⊂ ((X_{α,0}^{x} \ U)^{∗W_{0}} ∩ U_{y′,n})^{ΔW_{1}}.

Then since [y]_{G} is dense we may find y_{1} ∈ [y]_{G} ∩U′ ∩(U_{y′,n})^{ΔW_{1}}. Thus in particular we may find some g_{1} ∈ W_{1} with

g_{1} · y_{1} ∈ U_{y′,n}.

By assumption on U_{y′,n} we can find g_{2} ∈ V_{n} with

g_{2} · (g_{1} · y_{1}) ∈ U.

Thus we have

y_{1} ∈ U^{ΔV_{n}W_{1}}.

Thus by density of the orbit [x]_{G} in X_{α,0}^{x} we have that there is some x_{1} ∈ [x]_{G} with

x_{1} ∈ U′ ∩ U^{ΔV_{n}W_{1}}.

By assumptions on U′ we may find h_{1} ∈ W_{1} with

h_{1} · x_{1} ∈ (X_{α,0}^{x} \ U)^{∗W_{0}}.

But by assumption on x_{1} we must also have

h_{1} · x_{1} ∈ U′ ∩ U^{ΔV_{n}W_{1}h_{1}^{−1}} ⊂ U^{ΔV_{n}W_{1}W_{1}^{−1}},

with a contradiction to V_{n}W_{1}W_{1}^{−1} ⊂ W_{0}. (□Claim)

Now as in the proof of 9.5 we let O_{x,n,α} be the union of all U ∈ B_{α,0}^{x} which are such that whenever y_{0} ∈ U ∩ [y]_{G} and U′ is open with non-empty intersection with U then

∃g ∈ V_{n}(g · y_{0} ∈ U′).

Note that each O_{x,n,α} is dense in (X_{α,0}^{x}, τ_{α,0}^{x}).

Case(B2a): For some W_{0}, W_{1}, U_{y′,n} as in claim(1)

[x]_{G} ∩ ((X_{α,0}^{x} \ U)^{∗W_{0}} ∩ U_{y′,n})^{ΔW_{1}} ≠ ∅.

Then we let

B_{α+1}^{x} = B^{+}(B_{α,0}^{x}, B_{0}, {X \ O_{x,n,α} : n ∈ ℕ})

and we let τ_{α+1}^{x} be the topology generated by B_{α+1}^{x}. Thus by claim(1) there is certainly a τ_{α+1}^{x}-open set O with O ∩ [x]_{G} not relatively open in τ_{α,0}^{x}, and hence not open in τ_{α,0}^{x}.

Case(B2b): For every W_{0}, W_{1}, U_{y′,n} as in claim(1)

[x]_{G} ∩ ((X_{α,0}^{x} \ U)^{∗W_{0}} ∩ U_{y′,n})^{ΔW_{1}} = ∅.

Claim(2): For some n

x ∈ (X \ O_{x,n,α})^{ΔG}.

Proof of claim: Since [x]_{G} is included in a countable union of closed nowhere dense sets, may find n ∈ ℕ, x_{1} ∈ [x]_{G}, and W_{0}, W_{1} basic open neighborhoods of the identity such that

W_{1}^{−1} = W_{1},
V_{n}W_{1}W_{1}^{−1} ⊂ W_{0},

and W_{0}^{2} ∩ x_{1} is nowhere dense (in (X_{α,0}^{x}, τ_{α,0}^{x})).

But then if g_{1}·x_{1} ∈ U_{y′,n} any g_{1} ∈ W_{1} and U_{y′,n} as in the statement of claim(1), then by case assumption we have that for all open U meeting U_{y′,n} there is some ϵ ∈ W_{1} close to the identity with

ϵg_{1} · x ∉ (X_{α,0}^{x} \ U)^{∗W_{0}},
∴ ∃g_{0} ∈ W_{0}(g_{0}ϵg_{1} · x_{1} ∈ U).

Thus

W_{0}W_{1}^{2} · x_{1} ⊂ W_{0}^{2} · x_{1}

is somewhere dense, contradicting assumption on x_{1} and W_{0}. (□Claim) Claim(3): For no open W ⊂ G do we have that

[x]_{G} ∩ (X \ O_{x,n,α})^{∗W}

includes a relatively open (in [x]_{G}) non-empty set.

Proof of claim: Otherwise let O be such an open set in τ_{α,0}^{x} with

[x]_{G} ∩ O ⊂ (X \ O_{x,n,α})^{∗W} .

Then since [y]_{G} is dense we may find y_{1} ∈ O. By the definition of the O_{x,n,α} we clearly have

y_{1} ∈ O_{x,n,α}^{ΔW}.

By density of [x]_{G} we may find

x_{1} ∩ [x]_{G} ∩ O_{x,n,α}^{ΔW} ∩ O,

with a contradiction to the assumption on O. (□Claim) Then we let

B_{α+1}^{x} = B^{+}(B_{α,0}^{x}, B_{0}, {X \ O_{x,n,α} : n ∈ ℕ}).

By claim(3) we have that there is some open set U ∈ B_{α+1}^{x} whose intersection with [x]_{G} is not relatively open with respect to B_{α,0}^{x}, and thus certainly not with respect to B_{α}^{x}.

This completes the description of the transfinite changes in topologies. We need to see that this process terminates, and that around each orbit there are less than ω_{1} many changes of the topology.

Fix x ∈ X. For each α ∈ ω_{1} and U ∈ B_{α}^{x} we can let

W_{x,U} = {g ∈ G : g · x ∈ U},
A_{x,U} = {k ∈ ℕ : ∀g ∈ V̂_{k}(g · x ∈ U)}.

Since each (X_{α}^{x}, τ_{α}^{x}) is a Polish G-space, we have that each W_{x,U} is open in G. The A_{x,U} is simply a set of natural numbers describing W_{x,U} as a union of basic open sets. Finally for each α < ω_{1} let

π̂_{x} : ω_{1} → P_{ℵ_{0}}(2^{ℕ}),
π̂_{x} : α ↦ {A_{x,U} : U ∈ B_{α}^{x}}.

One can easily verify that this function is absolutely Δ ∼_{2}^{1} in the codes, α < β implies

π̂_{x}(α) ⊂ π̂_{x}(β),

and that as long as the process has not terminated, that is to say, as long as we are in case(B2) above, claim (3) implies

π̂_{x}(α) ≠ π̂_{x}(α + 1).

Thus by 9.4 we have some α at which π̂_{x}(α) equals π̂_{x}(α + 1) and we are in case(A) or case(B1). Case(A) for any x leaves us in alternative 2. of the theorem, so let us suppose instead that for every x we eventually come to rest in (B1).

For each x let α(x) be least α such that B_{α}^{x} = B_{α+1}^{x}. We always have that [x]_{G} is a G_{δ} in the topology generated by B_{α(x)}^{x}, and we wish to use its closure in that topology to represent the orbit by an HC set.

To this purpose we describe countable sets

A(x, α), A_{0}(x, α) ⊂ HC

for β ≤ α(x) and onto functions

F_{α}^{x} : A(x, α) ↠ B_{α}^{x}
F_{α,0}^{x} : A_{0}(x, α) ↠ B_{α,0}^{x}

for each α ≤ α(x).

The construction of F_{α+1}^{x} from F_{α,0}^{x} parallels 9.5. We build F_{α+1}^{x} as the union over i ∈ ℕ of F_{α,0,i}^{x} and A(β + 1, x) as the union of the A_{0}(x, β, i). We start with F_{α,0,0}^{x} = F_{α,0}^{x}.

F_{α,0,1}^{x}(⟨α, A_{0}(x, α), n⟩) = O_{x,n,α}.

Granted F_{α,0,i}^{x} at i > 0 and given l(0), l(1), ..., l(k), l, n ∈ ℕ and a, a′ ∈ A_{0}(x, α, i) with U, U′ ∈ B_{α,0}^{x} and

F_{α,0,i}^{x}(a) = U
F_{α,0,i}^{x}(a′) = U′

we let^{1}

F_{α,i+1}^{x}(⟨α, A_{0}(x, α, i), 0, a, a′, l(0), ..., l(k), l⟩) =

⟨a_{1}, a_{2}, ..., a_{n}⟩ = {{a_{1}, a_{2}, ..., a_{n}}, {a_{2}, ...., a_{n}}, ..., {a_{n}}}.

1Here the pairing function is from HC^{<ω} to HC – for instance

(U ∩ (X \ U′)^{∗W_{l(0)}} ∩ ... ∩ (X \ U′)^{∗W_{l(k)}})^{ΔW_{l}}
F_{α,i+1}^{x}(⟨α, A_{0}(x, α, i), 1, δ, δ′⟩) = U ∩ U′
F_{α,i+1}^{x}(⟨α, A_{0}(x, α, i), 2, δ, δ′⟩) = U ∪ U′.

We can let each A_{0}(x, α, i + 1) be A(x, α, i) along with the hereditarily countable sets above on which F_{α,i+1}^{x} was just defined.

The new ground is in defining F_{α,0}^{x} from F_{α}^{x}. We construct

F_{α,0,β}^{x} : A(x, α, β) ↠ B_{α}^{x} ∪ {U(φ_{β}^{τ_{α}^{x}}(x̂, U, V_{n})) : n ∈ ℕ, U ∈ B_{α}^{x}, β′ ≤ β, x̂ ∈ [x]_{G}}

by induction on β ≤ γ(x), with each F_{α,0,β}^{x} extending the function F_{α}^{x}. At the first level, if F_{α}^{x}(a) = U then

F_{α,0,0}^{x}(⟨α, A(x, α), a, n, {b : O(x̂, U, V_{n}) ∩ F_{α}^{x}(b) ≠ ∅}⟩) = U(φ_{0}^{τ_{α}^{x}}(x̂, U, V_{n}))

for any x̂ ∈ [x]_{G} and A(x, α, 0) is the union of A(x, α) and all the

⟨A(x, α), a, n, {b : O(x̂, U, V_{n}) ∩ F_{α}^{x}(b) ≠ ∅}⟩

indicated above. At successor stages of these definitions let

F_{α,0,β+1}^{x}(⟨α, A(x, α, β), a, n, {b : O(x̂, U, V_{n}) ∩ F_{α,0,β}^{x}(b) ≠ ∅}⟩) = U(φ_{β}^{τ_{α}^{x}}(x̂, U, V_{n}))

for any x̂ ∈ [x]_{G}; again the set A(x, α, β + 1) is the union of A(x, α) and all the sets of the form

⟨A(x, α, β), a, n, {b : O(x̂, U, V_{n}) ∩ F_{α,β}^{x}(b) ≠ ∅}⟩.

At limits λ ≤ γ(x)

F_{α,0,λ}^{x} = ⋃_{β<λ} F_{α,0,β}^{x}.

Finish with

F_{α,0}^{x} = F_{β,0,γ(x)}^{x}.

At the very final moment obtain

θ(x) = {a ∈ A(x, α(x)) : [x]_{G} ∩ F_{α(x)}^{x} ≠ ∅}.

Exercise 9.8. If X is a Polish S_{∞}-space, then x ↦ φ_{x}^{∗} (as defined in §6.1) is absolutely Δ ∼_{2}^{1}.

Exercise 9.9. If F is F_{2}-ergodic, then it is E_{S_{∞}}-ergodic, for any equivalence relation E_{S_{∞}} induced by the continuous action of S_{∞} on a Polish space. By induction on α, if π is a Baire measurable map satisfying x_{1}F x_{2} ⇒ π(x_{1})E_{S_{∞}}π(x_{2}) then x ↦ V_{α} ∩ φ_{π(x)}^{∗} is constant on a comeager set – where here V_{α} denotes the sets whose transitive closures have ∈-rank less than α. (Essentially this observation is implicit in Harvey Friedman’s proof that the cosets of the density ideal do not admit classification by countable structures.)

9.2. CARDINALITY IN L(ℝ) 169

9.2. Cardinality in L(ℝ)

The customary notion of cardinality is to define the cardinality of a set to be the least ordinal with which it may be placed in a bijection. In some respects the resulting theory is unsatisfying, since then there seems no agreed method to settle so simple a question as the cardinality of the continuum, and indeed for many of our most fundamental notions, such as ℝ, there are no definable well orderings.

There is another approach to cardinality – not so much a rival definition as an alternative concept. It is completely immune to the independence results associated with the continuum hypothesis.

Definition 9.10. Let A and B be sets in L(ℝ). We will say that the L(ℝ)- cardinality of A is less than that of B, written

|A|_{L(ℝ)} ≤ |B|_{L(ℝ)},

if there exists an injection i ∈ L(ℝ) with

i : A ↪ B.

The theory of L(ℝ) is canonical in the presence of sufficient large cardinal or determinacy assumptions, and there are no known interesting questions about its structure left unresolved by AD^{L(ℝ)}. Hence forth I will assume AD^{L(ℝ)}: the determinacy of all games of length ω on the integers are determined in L(ℝ); it implies a host of extraordinary regularity properties inside this model, including all sets Lebesgue measurable and the perfect set theorem for all subsets of ℝ.

Not only does L(ℝ) have an appealing internal structure, but it also includes anything we might want to consider reasonably definable. For instance, all Polish spaces up to homeomorphism, all Borel sets, all Σ ∼_{1}^{1}, C-measurable, and so forth.

Lemma 9.11. (folklore; AD^{L(ℝ)}) In L(ℝ) every non-empty Σ ∼_{1}^{2} subset of P(ℝ) has a Δ ∼_{1}^{2} member.

Proof. (Compare [65].) If Ψ is a Σ ∼_{1}^{2} property on sets of reals, and α is least such

L_{α}(ℝ) ⊧ ∃R(Ψ(R)),

then α is Σ ∼_{1}^{2} definable, L_{α}(ℝ) is the Σ ∼_{1}^{2} Skolem hull of the reals, and thus every element of L_{α}(ℝ) is a Σ ∼_{1}^{2} singleton in some real parameter.

Theorem 9.12. (Martin, Moschovakis, Steel; AD^{L(ℝ)}) Scale(Σ ∼_{1}^{2}) in L(ℝ).

Proof. See [65] or [66].

It is worth pointing out that for reasonably simple equivalence relations – certainly all Borel, but more generally all Σ ∼_{1}^{1}, and even all Δ ∼_{1}^{2} – the L(ℝ)-analog of the ≤_{B}-reducibility relation holds if and only if there is some L(ℝ) injection from the quotient space provided by the first into the quotient associated to the second.

Lemma 9.13. (AD^{L(ℝ)}) Let E, F be Δ ∼_{1}^{2} equivalence relations on 2^{ℕ} in L(ℝ). Then

|2^{ℕ}/E|_{L(ℝ)} ≤ |2^{ℕ}/F |_{L(ℝ)}

if and only if there is some function

θ : 2^{ℕ} → 2^{ℕ}

in L(ℝ) such that for all x, y ∈ 2^{ℕ}

xEy ⇔ (θ(x)F θ(y)).

Proof. The “if” direction of this lemma is trivial, so suppose instead that |2^{ℕ}/E|_{L(ℝ)} ≤ |2^{ℕ}/F |_{L(ℝ)}. Thus there is some S ⊂ 2^{ℕ} × 2^{ℕ} such that

∀x ∈ 2^{ℕ}∃y ∈ 2^{ℕ}((x, y) ∈ S),
∀x_{1}, x_{2} ∈ 2^{ℕ}∀y_{1}, y_{2} ∈ 2^{ℕ}((x_{1}, y_{1}), (x_{2}, y_{2}) ∈ S ⇒ (x_{1}Ex_{2} ⇔ y_{1}F y_{2})).

This is a Δ ∼_{1}^{2} property on S, so by 9.11 we may find some S_{0} ∈ Δ ∼_{1}^{2} as above, and then by 9.12 we can find a scale in L(ℝ) on S_{0}.

But since the scale property implies uniformization, we may actually find a function

θ : 2^{ℕ} → 2^{ℕ}

in L(ℝ) such that for all x ∈ 2^{ℕ}

(x, θ(x)) ∈ S_{0}

and thus

∀x_{1}, x_{2} ∈ 2^{ℕ}(x_{1}Ex_{2} ⇔ (θ(x_{1})F θ(x_{2}))).

Theorem 9.14. (AD^{L(ℝ)}) Let G be a Polish group and X a Polish G-space. The following are equivalent.

(I) E_{G}^{X} ≤_{aΔ_{2}^{1}} id(HC);

(II) there is no turbulent Polish G-space Y and a continuous G-embedding π : Y → X; (III) |X/G|_{L(ℝ)} ≤ |HC|_{L(ℝ)}.

Proof. We had the equivalence of (I) and (II) from 9.7, and (I) implying (III) is trivial. Thus it suffices to show that if Y is a turbulent Polish G-space then there is no injection i in L(ℝ) from X/G to HC.

But here the set of such injections is Σ ∼_{1}^{2}, and so by 9.11 if there is such an i then we may assume it to be Σ ∼_{1}^{2} in the codes. Next by 9.12 we can obtain an honest to goodness function θ ∈ L(ℝ) with

θ : X → Mod(L(∈))

so that for all x_{1}, x_{2} ∈ X

x_{1}E_{G}^{X}x_{2} ⇔ θ(x_{1}) ≅ θ(x_{2}).

This contradicts 3.18 since all functions in L(ℝ) are Baire measurable.

Theorem 9.15. (AD^{L(ℝ)}) Let G be a GE group and X a Polish G-space. Then the following are equivalent:

(I) E_{G}^{X} admits classification by countable models;

(II) there is a countable language L with E_{G}^{X} ≤_{F B}≅|_{Mod(L)};

(III) x ↦ φ_{x} provides a complete invariant for the orbit equivalence relation; (IV) there is no turbulent Polish G-space Y and continuous G-embedding from Y to X; (V) |X/G|_{L(ℝ)} ≤ |HC|_{L(ℝ)}.

Proof. 9.14 and 6.30.

Lemma 9.16. (AD^{L(ℝ)}) In L(ℝ) all well ordered intersections of comeager sets are comeager.

Proof. By Kuratowski-Ulam, and [67] for all sets in L(ℝ) having Baire property.

Notation 9.17. Let L be a language and κ an ordinal. S_{κ} denotes the permutation group on κ and Mod(L, κ) denotes the collection of all models whose underlying set is κ. We may write

E ≤_{L(ℝ)}≅|_{Mod(L,κ)}

if there is a function θ in L(ℝ) so that for all x_{1}, x_{2}

x_{1}Ex_{2} ⇔ L(ℝ) ⊧ (θ(x_{1}) ≅ θ(x_{2})).

Here we specifically ask that the isomorphism between the models exist in L(ℝ).

Theorem 9.18. (AD^{L(ℝ)}) Let G be a Polish group and X be a Polish G-space. If for some well ordered language L and ordinal κ

E_{G}^{X} ≤_{L(ℝ)}≅|_{Mod(L,κ)}

then the action is not turbulent.

Proof. Suppose instead that X is turbulent. Work in L(ℝ).

The proof is akin to 3.18, only replacing the countability of ℕ with the well orderability of L and κ and the use of 9.16. In the case of 3.18 we can simply enumerate the basic open sets in S_{∞} in search of the sequence θ(x_{i}) → g_{∞} · θ(x). Here we need to stitch together a well ordered collection of basic open sets, meeting them not one after another according to a fixed enumeration, but strategically, in a construction that remembers ideas from forcing and applications of MA(ℵ_{1}).

Reintroduce all our usual notation for the group action. Let B = {U_{l} : l ∈ ℕ} be a countable basis for X, B_{0} for G, G_{0} = {g_{j} : j ∈ ℕ} a countable dense subgroup of G, (V_{n})_{n∈ℕ} a countable basis at 1_{G} each V_{n} = V_{n}^{−1}. Assume moreover that each V_{n} ∈ B_{0} and that B_{0} is closed under left and right translation by elements in G_{0}.

By the basis theorem at 9.11 we can assume that there is a Δ ∼_{1}^{2} reduction

θ : X → Mod(L, κ)

for κ < δ_{∼}_{1}^{2} so that for all g ∈ G

θ(x) ≅ θ(g · x).

Then by the “coding lemma” of [67] we may code each element of S_{κ} by a real, and then gather from 9.12 that there is a function

Γ_{0} : X × G → S_{κ}

so that for all (x, g) ∈ X × G

Γ_{0}(x, g) : θ(x) ≅ θ(g · x).

For each basic open U ⊂ X and V ⊂ G let A_{0}(U, V) be the set of α < κ such that there exists some β < κ at which

∀^{∗}x ∈ U∀^{∗}g ∈ V (Γ_{0}(x, g)(α) = β).

Note that U′ ⊂ U and V′ ⊂ V implies A_{0}(U′, V′) ⊃ A_{0}(U, V).

Claim(1): Let U, U′ ⊂ X be basic open and n ∈ ℕ with V_{n} · U′ ⊂ U, h ∈ G_{0}. Then for a comeager set of x ∈ U′ and g ∈ V_{n} there exists

π : θ(x) ≅ θ(g · x)
π|_{A_{0}(U,hV_{n})} = id

(i.e. π(α) = α all α ∈ A_{0}(U, hV_{n})).

Proof of claim. Using 9.16 we have that there is some

σ : A_{0}(U, hV_{n}) → κ

for whom ∀^{∗}x ∈ U∀^{∗}g ∈ hV_{n}

Γ_{0}(x, g)|_{A_{0}(U,hV_{n})} = σ.

For any sufficiently generic x ∈ U′ and g ∈ V_{n} we may find h′ ∈ G so that h′h, h′hg ∈ hV_{n} and

Γ_{0}(x, h′hg)|_{A_{0}(U,hV_{n})} = σ
Γ_{0}(g · x, h′hg)|_{A_{0}(U,hV_{n})} = σ.

Then

Γ_{0}(g · x, h′h) : g · x ↦ (h′hg) · x,
Γ_{0}(x, h′hg) : x ↦ (h′hg) · x,

and so

(Γ_{0}(g · x, h′hg))^{−1}Γ_{0}(x, h′hg)

is as needed. (Claim(1)□)

Claim(2): For all α < κ the union of all U ∈ B such that for some n ∈ ℕ and h ∈ G_{0} we have

α ∈ A_{0}(U, hV_{n}).

Proof of claim. Clearly this set is open. As for density, at each β < κ let

D_{β} = {(x, g) : (Γ_{0}(x, g))(α) = β}.

Then by 9.16 and separability of X × G there is some countable set (β_{i})_{i∈ℕ} such that

⋃_{i∈ℕ} D_{β_{i}}

is comeager in X × G. Thus by Kuratowski-Ulam

∀^{∗}x ∈ X∀^{∗}g ∈ G((g, x) ∈ ⋃_{i∈ℕ} D_{β_{i}}).

For each let

D_{β_{i},x} = {g ∈ G : (g, x) ∈ D_{β_{i}}}.

By all sets having Baire property

∀^{∗}x ∈ X∀^{∗}h ∈ G∃n ∈ ℕ∃i ∈ ℕ(∀^{∗}g ∈ V_{n}((g, h · x) ∈ D_{β_{i}}))
∴ ∀^{∗}x ∈ X∃n ∈ ℕ∃i ∈ ℕ(∀^{∗}g ∈ V_{n}((g, x) ∈ D_{β_{i}}))

by Kuratowski-Ulam.

For each x in this comeager set let (n_{x}, i_{x}) be the least pair in ℕ × ℕ under the lexicographic order such that

∀^{∗}g ∈ V_{n}((g, x) ∈ D_{β_{i}}).

Since x ↦ (n_{x}, i_{x}) is continuous on a comeager set we can find a collection of basic open subsets U whose union is dense in X and such that on any U ∈ U the function x ↦ (n_{x}, i_{x}) is constant on a relatively comeager subset. (Claim(2)□)

Thus in particular the claim implies that for each α < κ the union of all open U′ ⊂ X such that there exists U ∈ B, n ∈ ℕ, and h ∈ G_{0}, with

V_{n} · U′ ⊂ U,

and

α ∈ A_{0}(U, hV_{n})

will be open dense.

Claim(3): We can find C′_{0} ⊂ X × G comeager

Γ_{1} : C′_{0} × ℕ^{4} → S_{κ}

with

∀(x, g) ∈ C′_{0}(Γ_{1}(x, g, (l, k, n, j)) : θ(x) ≅ θ(g · x))

and such that for all (x, g) ∈ C′_{0} and l, k, n, j ∈ ℕ, if x ∈ U_{k},

V_{n} · U_{k} ⊂ U_{l}
g · x ∈ O(x, U_{k}, V_{n})

then

Γ_{1}(x, g, (l, k, n, j))|_{A_{0}(U_{l},h_{j}V_{n})} = id|_{A_{0}(U_{l},h_{j}V_{n})}.

Proof of claim. For sufficiently generic x, g with g ∈ V_{n} we can certainly find

σ : θ(x) ≅ θ(g · x)

with

σ|_{A_{0}(U_{l},h_{j}V_{n})} = id|_{A_{0}(U_{l},h_{j}V_{n})}.

More generally given x ∈ U_{n} and g ∈ G sufficiently generic with

g · x ∈ O(x, U_{k}, V_{n})

we may find g_{0}, g_{1}, ..., g_{m} ∈ V_{n} so that for

x_{0} = x,
x_{i+1} = g_{i} · x_{i},

we have each x_{i} ∈ U_{k} and each (x_{i}, g_{i}) meeting the comeager set shown to exist in claim(1) for A_{0}(U_{l}, h_{j}V_{n}). Thus there is at each i some

σ_{i} : θ(x_{i}) ≅ θ(x_{i+1})

with

σ_{i}|_{A_{0}(U_{l},h_{j}V_{n})} = id_{A_{0}(U_{l},h_{j}V_{n})},

and we may find σ as their composition.

Uniformization for Σ ∼_{1}^{2} finds some Γ_{1} producing σ as a function of x and g. (Claim(3)□)

By claim (2) and 9.16 we can find comeager C_{0} ⊂ C′_{0} so that for all (x, g) ∈ C_{0} and α < κ there is some U, U′ ∈ B, n ∈ ℕ, h ∈ G_{0}

x ∈ U′
V_{n} · U′ ⊂ U
α ∈ A_{0}(U, hV).

Repeat: Let A_{1}(U, V, (l, k, n, j)) be the set of α < κ for which there is some β with

∀^{∗}x ∈ U∀^{∗}g ∈ V (Γ_{1}(x, g, (l, k, n, j))(α) = β).

Iterating we obtain for each i ∈ ℕ some comeager

C_{i} ⊂ X × G
Γ_{i+1} : C_{i} × (ℕ^{4})^{i+1} → S_{κ}

so that all (x, g) ∈ C_{i} and m⃗ = ((l(0), k(0), n(0), j(0)), ..., (l(i), k(i), n(i), j(i)) ∈ (ℕ^{4})^{i+1} we have that if

x ∈ U_{k(i)}
g · x ∈ O(x, U_{k(i)}, V_{n(i)})
V_{n(i)} · U_{k(i)} ⊂ U_{l(i)}

then

Γ_{i+1}(x, g, m⃗)|_{A_{i}(U_{l(i)},h_{j(i)}V_{n(i)},m⃗|_{i})} = id;

and for all α < κ, (x, g) ∈ C_{i}, m⃗ ∈ (ℕ^{4})^{i}, there are U, U′ ∈ B, n ∈ ℕ, h ∈ G_{0}

x ∈ U′
V_{n} · U′ ⊂ U
α ∈ A_{i}(U, hV, m⃗).

Then define A_{i+1}(U, V, m⃗) be the set of α so that for some β

∀^{∗}x ∈ U∀^{∗}g ∈ V ((Γ_{i+1}(x, g, m⃗))(α) = β).
Claim(4): For m⃗ ∈ (ℕ^{4})^{i+1}
U ⊂ U_{k(i)},
V ⊂ V_{n(i)}

we have

A_{i+1}(U, V, m⃗) ⊂ A_{i}(U_{l(i)}, h_{j(i)}V_{n(i)}, m⃗|_{i}).

Proof of claim: A tautology given the above specifications on Γ_{i+1}. (Claim(4)□) Use 9.16 to produce a comeager set C on which θ is continuous in the sense

that for any R ∈ L and α⃗ ∈ κ^{<ℕ}

θ^{−1}[{M : M ⊧ R(α⃗)}] ∩ C

is relatively clopen. Let C_{∞} be the set of x ∈ C so that

∀g ∈ G_{0}∀i ∈ ℕ((x, g) ∈ C_{i}, g · x ∈ C).

It suffices to pick x and y in C_{∞} and demonstrate an isomorphism between θ(x) and θ(y).

For this, let Z(x, y) be the space of all infinite sequences

((h_{i})_{i∈ℕ}, (h′_{i})_{i∈ℕ}, ((l(i), k(i), n(i), j(i))_{i∈ℕ})

in G_{0}^{ℕ} × G_{0}^{ℕ} × (ℕ^{4})^{ℕ} such that for

x = x_{0}, x_{1}, x_{2}, ...

and

y_{0} = y, y_{1}, y_{2}, ...

defined by

x_{i+1} = h_{i} · x_{i}

9.2. CARDINALITY IN L(ℝ) 175

and

y_{i+1} = h′_{i} · y_{i}

we have:

(i) h_{i} ≈_{k(2i),n(2i)}^{x_{i}} 1_{G} (and therefore x_{i+1} ∈ O(x_{i}, U_{k(2i)}, V_{n(2i)}));

(ii) h′_{i} ≈_{k(2i+1),n(2i+1)}^{y_{i}} 1_{G} (and therefore y_{i+1} ∈ O(y_{i}, U_{k(2i+1)}, V_{n(2i+1)}));

(iii) U_{k(i+1)} ⊂ U_{l(i+1)} ⊂ U_{k(i)} ⊂ U_{l(i)};
(iv) x_{i} ∈ U_{k(2i)};
(v) y_{i} ∈ U_{k(2i+1)};
(vi) V_{n(i+1)} ⊂ V_{n(i)};
(vii) O(y_{i}, U_{k(2i+1)}, V_{n(2i+1)}) is dense in U_{l(2i+2)};
(viii) O(x_{i}, U_{k(2i)}, V_{n(2i)}) is dense in U_{l(2i+1)};
(ix) V_{n(i)} · U_{k(i)} ⊂ U_{l(i)}.

The definition indicates Z(x, y) to be closed in the space G_{0}^{ℕ} × G_{0}^{ℕ} × (ℕ^{4})^{ℕ} and so Polish in the product topology (where here we consider G_{0} as a discrete space). We need a well ordered collection of open dense subsets, parallel to 3.18:

For R in L and α⃗ ∈ κ^{<ℕ} let D_{R,α⃗} be the set of (h⃗, h⃗′, m⃗) ∈ Z(x, y) so that for some i either

∀^{∗}z ∈ U_{l(i)}(θ(z) ⊧ R(α⃗))

or

∀^{∗}z ∈ U_{l(i)}(θ(z) ⊧ ¬R(α⃗)).

For α < κ let D_{θ,α}^{0} be the set of (h⃗, h⃗′, m⃗) ∈ Z(x, y) so that at some i if we let β_{i,α} =_{df} (Γ_{2i−2}(x_{i−1}, h_{i−1}, m⃗|_{2i−2})) ∘ (Γ_{2i−4}(x_{i−2}, h_{i−2}, m⃗|_{2i−4})) ∘ ...(Γ_{0}(x_{0}, h_{0}))(α) then

β_{i,α}, α ∈ A_{2i−1}(x_{i}, U_{l(2i)}, g_{j(2i)}V_{n(2i)}, m⃗|_{2i});

note that from this, (i), (ix), and the assumptions on the Γ_{i} functions give that

(Γ_{2i}(x_{i}, h_{i}))(α) = α,

(Γ_{2i}(x_{i}, h_{i}))(β_{i,α}) = β_{i,α}.

Similarly let D_{θ,α}^{1} be the set of (h⃗, h⃗′, m⃗) ∈ Z(x, y) for which there is some i such that if γ_{i,α} =_{df} (Γ_{2i−1}(y_{i−1}, h′_{i−1}, m⃗|_{2i−1})) ∘ (Γ_{2i−3}(y_{i−2}, h′_{i−2}, m⃗|_{2i−3})) ∘ ...(Γ_{0}(y_{0}, h′_{0}))(α) then

γ_{i,α}, α ∈ A_{2i}(y_{i}, U_{l(2i+1)}, g_{j(2i+1)}V_{n(2i+1)}, m⃗|_{2i+1});

again from this, (ii), (ix), and the assumptions on the Γ_{i}’s we obtain

(Γ_{2i+1}(y_{i}, h′_{i}))(α) = α,

(Γ_{2i+1}(y_{i}, h′_{i}))(γ_{i,α}) = γ_{i,α}.

Once we obtain some i with

β ∈ A_{2i−1}(x_{i}, U_{l(2i)}, g_{j(2i)}V_{n(2i)}, m⃗|_{2i})

or γ with

γ ∈ A_{2i}(y_{i}, U_{l(2i+1)}, g_{j(2i+1)}V_{n(2i+1)}, m⃗|_{2i+1})

then for all i′ > i

β ∈ A_{2i′−1}(x_{i′}, U_{l(2i′)}, g_{j(2i′)}V_{n(2i′)}, m⃗|_{2i′})

or

γ ∈ A_{2i′}(y_{i′}, U_{l(2i′+1)}, g_{j(2i+1)}(2V_{n(2i′+1)}, m⃗|_{2i′+1})

respectively by (iii), (vi), and claim(4). Thus we respectively would have either

(Γ_{2i′}(x_{i′}, h_{i′}))(α) = α,
(Γ_{2i′}(x_{i′}, h_{i′}))(β_{i,α}) = β_{i,α}

or

(Γ_{2i′+1}(y_{i′}, h′_{i′}))(α) = α,
(Γ_{2i′+1}(y_{i′}, h′_{i′}))(γ_{i,α}) = γ_{i,α}.

Each D_{R,α⃗} is dense by assumption of θ continuous on C. All the D_{θ,α}^{0} and D_{θ,α}^{1} are dense by assumption on the C_{i}’s.

By 9.16 obtain some sequence (h⃗, h⃗′, m⃗) ∈ Z(x, y) meeting all these comeager sets. Then if we set

σ_{i} = Γ_{2i}(x_{i}, h_{i}, m⃗|_{2i})
σ′_{i} = Γ_{2i+1}(y_{i}, h′_{i}, m⃗|_{2i+1})

and

M_{i} = θ(x_{i})
N_{i} = θ(y_{i})

then by the definitions of the Γ_{i} functions

σ_{i}(M_{i}) = M_{i+1}
σ′_{i}(N_{i}) = N_{i+1}.

By meeting all the D_{R,α⃗} sets and the fact that the x_{i}’s and y_{i}’s stay inside the comeager set of points on which θ is continuous we have some M_{∞} with

M_{i} → M_{∞}
N_{i} → M_{∞}.

From the D_{θ,α}^{0} and D_{θ,α}^{1} there are σ_{∞} ∈ S_{κ} and σ′_{∞} ∈ S_{κ} with

σ_{i} ∘ σ_{i−1} ∘ ...σ_{0} → σ_{∞}
σ′_{i} ∘ σ′_{i−1} ∘ ...σ′_{0} → σ′_{∞}.

Then

σ_{∞}(M_{0}) = σ′_{∞}(N_{0}).

Corollary 9.19. Let G be a Polish group, X a Polish G-space, κ an ordinal, and L a well ordered language. If

E_{G}^{X} ≤_{L(ℝ)}≅ |_{Mod(L,κ)}

then for some countable language L′

E_{G}^{X} ≤_{L(ℝ)}≅ |_{Mod(L′)};

in fact for suitably chosen L′

E_{G}^{X} ≤_{aΔ_{2}^{1}} ≅ |_{Mod(L′)}.

CHAPTER 10 Looking ahead

I will finish with some questions and conjectures, everyone of which I consider interesting and important to this direction. Several – such as 10.4, 10.5, and 10.6 – I am inclined to consider difficult.

In some ways the main results of this manuscript could be a considered as a reaction to the failure to find an analogue of Harrington-Kechris-Louveau for the notion of turbulence. Here there are two sibling theorems, proved by similar ideas.

Theorem 10.1. Let G be a Polish group and X a Polish G-space. Then the following are equivalent:

(I) E_{G}^{X} is reducible to id(2^{<ω_{1}}) by a function that is universally Baire measurable in the codes (in the sense 3.12); (II) there is no properly generically ergodic Polish G-space Y and continuous G-embedding π : Y ↪ X.

Theorem 10.2. Let G be a Polish group and X a Polish G-space. Then the following are equivalent:

(I) E_{G}^{X} is reducible to id(HC) by a function that is universally Baire measurable in the codes (compare 6.18); (II) there is no turbulent Polish G-space Y and continuous G-embedding π : Y ↪ X.

In both cases we thus have that in some sense the presence of a certain dynamical property is a necessary and sufficient condition for non-classifiability. For actually calculating the classification difficult of a given problem the conditions “properly generically ergodic” and “turbulent” almost always appear at least implicitly through some reduction, and 10.1 and 10.2 seem to provide an explanation for this fact.

It should be admitted that the proofs of both these results are completely abstract and use non-trivial set theoretical machinery. One simply sets into place a transfinite process of changing the topology around a point in the space, and shows that the operation must terminate before ω_{1} steps for any orbit, and it can only terminate by either displaying a properly generically ergodic or turbulent space embedded into X or by producing an invariant of the required type.

The main defect in these arguments is that they give no further information when E_{G}^{X} is Borel. For instance, by combining 10.1 with [30] one has:

Corollary 10.3. Let G be a Polish group and X a Polish G-space with E_{G}^{X} Borel. Then the following are equivalent:

(I) E_{G}^{X} ≤_{B} id(ℝ);

(II) there is no properly generically ergodic Polish G-space Y and continuous G-embedding π : Y ↪ X.

But the point is that the proof of 10.1 does not give this result on its own. Similarly in the proof of 10.2 one obtains no benefit by assuming E_{G}^{X} is Borel. Thus we are left with questions such as:

Question 10.4. Let G be a Polish group and X a Polish G-space with E_{G}^{X} Borel. Must one of the following hold?

1. E_{G}^{X} is reducible to id(HC) by a function that is Borel in the codes;

2. there is a turbulent Polish G-space Y and continuous G-embedding π : Y ↪ X.

Indeed one can even ask:

Question 10.5. Let G be a Polish group and X a Polish G-space. Must one of the following hold?

1. E_{G}^{X} ≤_{F B} E_{S_{∞}}^{Y} for some Polish S_{∞}-space Y;

2. there is a turbulent Polish G-space Y and continuous G-embedding π : Y ↪ X.

In the context of the smooth/non-smooth distinction, it is exactly understood which Polish groups only give rise to smooth orbit equivalence relations: Clearly the orbits under the continuous action by a compact group are all closed, and following say proofs of §7.1 we have reducibility to the identity relation on ℝ; conversely Solecki showed that any non-compact Polish group can act continuously on a Polish space with non-smooth orbit equivalence relation.

The situation for classification by countable models is far less clear. From [4] and §7.4 these include S_{∞}, its closed subgroups, the ℵ_{0} products of locally compact groups, and all their closed subgroups. We might then hope that this class of groups allowing countable models for invariants should be in some natural way generated from these examples, but in fact I think there is at least one more kind of group that needs to be considered:

Conjecture 10.6. Whenever Hom^{+}([0, 1]) acts continuously on a Polish space the orbit equivalence relation admits classification by countable models.

Apart from the above problems – all of which are more or less directly connected with trying to understand which equivalence relations may be simply subsumed under the study of countable models – we can also consider a number of parallel issues. Somewhat more general than the equivalence relations induced by S_{∞} are those arising from continuous actions of the unitary group, U_{∞}. Perhaps the most interesting problem in this regard is a hopelessly vague one: Does there exist a canonical transfinite sequence of invariants that serve U_{∞} as well as the φ_{α} invariants from §6.1? It is hard to make this question more mathematically exact, but one way might be to ask if there is an analogue 6.11 assigning a privileged kind of invariant when the orbit equivalence relation has low Borel complexity.

Question 10.7. Let X be a Polish U_{∞}-space with E_{U_{∞}}^{X} in F_{σδ}. Is E_{U_{∞}}^{X} necessarily Borel reducible to probability measures on the unit interval considered up to absolute continuity? That is to say must there exist a Borel assignment x ↦ μ_{x} such that x_{0}E_{U_{∞}}^{X}x_{1} if and only if μ_{x_{0}} and μ_{x_{1}} have the same null sets?

It is known from the spectral theorem as found in [10] or [83] that we do indeed obtain such a reduction for the equivalence relation of conjugacy in the unitary group U_{∞}.

Of course there are many problems with this flavor, and ultimately one can ask about invariants for arbitrary Polish groups. It is not clear what form such an analysis might take, so we might propose the following test. The case for G = S_{∞} was established by Harvey Friedman using the Scott analysis:

Question 10.8. For which Polish groups G do we have the following: Whenever X is a Polish G-space and E a Borel equivalence relation with E ≤_{B} E_{G}^{X} there must exist a Polish G-space Y with E ≤_{B} E_{G}^{Y} and E_{G}^{Y} Borel?

As far as I am aware, this could be all Polish groups. The proof of the affirmative answer for G = S_{∞} is given in [19].

We can also ask about a structure for reduction to group actions. Very tepidly and on the basis of wafer thin evidence the authors of [39] conjecture a positive answer to:

Question 10.9. For E a Borel equivalence relation must one of the following hold?

1. E ≤_{B} E_{G}^{X} for some Polish group G and G-space X; or

2. E_{1} ≤_{B} E.

From 8.2 at most one. In a restricted case [77] upholds this:

Theorem 10.10. (Solecki) For I a Borel ideal on 2^{ℕ} exactly one of the following:

1. 2^{ℕ}/I ≤_{B} E_{G}^{X} for some Polish group G and Polish G-space X; or

2. E_{1} ≤_{B} 2^{ℕ}/I.

Even in specific situations the conjecture is not known to hold, in as much as frequently it is not clear when we can reduce to a group action. For instance:

Question 10.11. In some suitable standard Borel structure, is biholomorphism on complex manifolds ≤_{B} E_{G}^{X} for some Polish group G and some Polish G-space X?

There are also two incidental problems which appeared in the course of chapter 7:

Question 10.12. (Kechris; [49]) If G is a Polish group for which E_{G}^{X} is always Borel reducible to a countable equivalence relation, must it be the case that G is locally compact?

Question 10.13. (Becker; [3]) Are the CLI groups the only ones with the weak Glimm-Effros property?

180 10. LOOKING AHEAD

APPENDIX A Ordinals

Ordinals arise from the need to keep counting through infinitely many stages. For instance, if we define the Σ ∼_{1}^{0} subsets of ℝ to be the open sets and Π ∼_{1}^{0} to be the closed, and more generally Σ ∼_{n+1}^{0} to be the countable unions of Π ∼_{n}^{0} and Π ∼_{n+1}^{0} to be the complements of Σ ∼_{n+1}^{0} then we obtain an initial segment of the Borel sets consisting of those with finite rank. The issue here is that not all Borel sets are in these classes. A Borel set of the form

B = ⋃_{n∈ℕ} B_{n}

with each B_{n} ∈ Σ ∼_{n}^{0} may provide a counterexample.

So we are led to a notation for the infinite rank Borel sets. The first infinite ordinal is ω and we can let Σ ∼_{ω}^{0} be the countable unions of finite rank Borel sets, Π ∼_{ω}^{0} their complements, Σ ∼_{ω+1}^{0} to be the countable unions of Π ∼_{ω}^{0}, and so on.

The above provides the motivation. The formal definition is:

Definition A.1. A set α is an ordinal if (i) α is transitive (so for all β ∈ α and γ ∈ β we have γ ∈ α); (ii) α is linearly ordered by the ∈-relation (so for all β, γ ∈ α, either β ∈ γ, or γ ∈ β, or γ = β).

α is a successor ordinal if it has a largest element, and otherwise it is a limit.

Example A.2. The first ordinal is the empty set, ∅, which we may identify with 0. Then

1 = {0}

the set whose only member is 0(=_{df} ∅). Then

2 = {0, 1}

and more generally

n + 1 = {0, 1, 2, ..., n}.

The first infinite ordinal is

ω = {0, 1, 2, 3, ...}

and we keep counting with

ω + 1 = {0, 1, 2, 3, ...ω},
ω + 2 = {0, 1, 2, 3, ...ω, ω + 1},

reaching the next limit at

ω + ω = {0, 1, 2, 3, ...ω, ω + 1, ω + 2, ...}.

Strictly speaking 0 is a limit ordinal, but it is frequently put in a class by itself. In any case, with the exception of this minor point, we can divide the ordinals into the classes of successor and limit.

At once from the definition at A.1 we have that any member of an ordinal is again an ordinal.

Definition A.3. A set A is said to be well ordered by a relation < if (i) < is well founded, in the sense that every non-empty subset of A has a <-least element:

∀B ⊂ A(B ≠ ∅ ⇒ ∃b ∈ B(∀a ∈ A(a < b ⇒ a ∉ B)));

(ii) A is linearly ordered by the <-relation.

Two immediate consequences of the usual formalizations of mathematical reasoning are:

Theorem A.4. Every ordinal is well ordered by ∈.

Theorem A.5. If < well orders the set A then there is some ordinal α so that (A, <) and (α, ∈) are isomorphic as linear orderings.

For our purposes we may as well take A.4 and A.5 as axioms.

A.4 provides the foundation for arguments by transfinite induction. Suppose C ⊂ α includes 0, and for all β ∈ α it includes β + 1, and whenever λ ∈ α is a limit ordinal with λ ⊂ C we have λ ∈ C, then we may conclude that C equals α: Otherwise there will be some ∈-least γ ∈ (α \ C) by the assumption ∈ provides a well ordering on α; but then this least γ can be neither successor nor limit by the assumptions on C.

Definition A.6. A set is countable if there is an onto map

π : ℕ ↠ A.

ω_{1} is the first uncountable ordinal.

Thus we allow that finite sets are countable. A set is said to have cardinality ℵ_{0} if it is infinite and countable, and we express this with the notation

|A| = ℵ_{0}.

Since there is a bijection between ℕ and ℕ × ℕ, countable unions of countable sets are countable.

Lemma A.7.

Borel = ⋃_{α∈ω_{1}} Π ∼_{α}^{0}

Proof. ⊃ is clear from the definition of Borel as the smallest superset of open forming a σ-algebra.

For the other direction we have that open sets are Π ∼_{2}^{0} and so should only be concerned with showing ⋃_{α∈ω_{1}} Π ∼_{α}^{0} forms a σ-algebra. Suppose that B_{n} ∈ Π ∼_{α(n)}^{0} for each n ∈ ℕ. Note that since ω_{1} is not the countable union of countable sets there will be some δ < ω_{1} with each α(n) < δ. But then

⋃_{n∈ℕ} B_{n} ∈ Σ ∼_{δ}^{0} ⊂ Π ∼_{δ+1}^{0}.

A. ORDINALS 183

For a thorough treatment see [45] or [68].

184 A. ORDINALS

APPENDIX B Notation

Notation B.1. Sets ∅ is the empty set, uniquely defined by the description that it has no members.

For A, B both sets, A ⊂ B indicates that A is included in B; A ⊂ B is not taken to mean that A is necessarily unequal to B, so for instance it is true that

∅ ⊂ ∅.

A \ B is the set of all points in A and not in B. A ∪ B is the set of points in either A or B and A ∩ B is the set of points in both. AΔB indicates the symmetric difference of these sets:

AΔB = (A \ B) ∪ (B \ A).

For A some set and Q(·) some property, we use

{x ∈ A : Q(x)}

and

{x ∈ A| Q(x)}

interchangeably to mean the set of all points in A with the property Q. This allows some deviations, for instance if f is a function, then

{f(x) : Q(x)}

would indicate the set of all images under f of x’s which have Q. In the case of functions it would look strange to write

{f : X → Y : Q(f)},

and so we use

{f : X → Y | Q(f)}

to indicate the set of functions from X to Y which have property Q.

P(A) is the set of all subsets of A.

Notation B.2. Sequences A sequence – that is to say a function from ℕ or some finite initial segment of ℕ – is commonly denoted by vector notation, as in x⃗; here the reader should expect that the nth term of this sequence is x_{n}. Alternative notations for sequences include (·, ·, ...) and ⟨·, ·, ...⟩. For instance the sequence (a_{0}, a_{1}, ..., a_{k−1}) has k elements and its second term is a_{1}.

For s, t finite sequences, we denote the concatenation by s^{⌢}t. So for l the length of s and i < l we have

(s^{⌢}t)(i) = s(i),

whilst for j ≥ l we let

(s^{⌢}t)(j) = t(j − l).

There is a variant for the case that one of the sequences has length one, where we write s^{⌢}a or a^{⌢}s instead of s^{⌢}⟨a⟩ or ⟨a⟩^{⌢}s. Thus for l = length of s,

(s^{⌢}a)(i) = s(i), i < l,
(s^{⌢}a)(l) = a,
(a^{⌢}s)(0) = a,
(a^{⌢}s)(i + 1) = s(i), i < l.

We also write (x_{n})_{n∈ℕ} to indicate the sequence (x_{0}, x_{1}, ...).

For a set A, we let A^{ℕ} denote the collection of all infinite sequences from A and A^{<ℕ} denote the collection of all finite sequences.

Given the identification of each natural number with its predecessors, we thus have that 2^{n} equals the set of functions from {0, 1, ..., n − 1} to {0, 1}. So in this context, 2^{1} does not equal 2, but rather the set {v_{0}, v_{1}}, where v_{0} is the sequence of length 1 whose only term is 0 and v_{1} is the sequence whose only term is 1.

Of course this may seem an insane convention, but it makes for compacting expressions which would otherwise be verbose. In context it will never create confusion, but the reader should note that 2^{−n} always equals the inverse of 2 to the nth power. In particular 2^{−1} equals

\frac{1}{2},

and not some exotic attempt to invert the set {v_{0}, v_{1}} from above.

Notation B.3. Functions Given a set A and a function f with domain A and a subset B ⊂ A, we use f|_{B} to denote the restriction of the function f to B. In the special case that

f : ℕ → C

is a function from ℕ to some set C we use f|_{n} to denote the restriction of f to the set {0, 1, ..., n − 1}. Since we understand ℕ = {0, 1, 2, ...} and each n = {0, 1, ..., n − 1}, this is entirely consistent. A^{B} is used to indicate the set of all functions from B to A.

In general f[A] = {f(x) : x ∈ A} whenever the set A is a subset of the domain of f, and f^{−1}[B] = {x : f(x) ∈ B} when B is a subset of the range.

f : X → Y

indicates that f is a function with domain X and range Y. An expression such as

f : x ↦ \frac{x}{5} + 7

indicates that f is the function which takes x, divides by 5, adds seven, and spits out

\frac{x}{5} + 7.

If X is a set and A ⊂ X a subset, then we define the characteristic function of A on X by

χ_{A} : X → {0, 1},
χ_{A}(x) = 1 if x ∈ A,
χ_{A}(x) = 0 if x ∉ A.

If X × Y is a product of two spaces and f : X × Y → B is a function whose domain is the product, we prefer the informal f(x, y) to denote the value of f applied to the pair (x, y) ∈ X ×Y, rather than the fussier but more correct f((x, y)). Given x ∈ X we let

f(x, ·) : Y → B
y ↦ f(x, y).

If A is a set, then (x_{a})_{a∈A} indicates the function with domain A which assumes the value x_{a} for any a ∈ A. So in particular in the case that A = ℕ this leads to one of the usual notations, (x_{n})_{n∈ℕ}, for a sequence.

Notation B.4. Topological notions We customarily identify a topological space with its underlying set.

If X is a topological space and A ⊂ X, then \overline{A} denotes its closure, A^{o} denotes its interior (that is the largest open set included in A), and

∂A = \overline{A} ∩ \overline{X \ A}

indicates the boundary.

In the case that X is a metric space and d is the metric, we let

d(B) = sup {d(b_{1}, b_{2}) : b_{1}, b_{2} ∈ B}.

For b ∈ X

d(b, B) = inf {d(b, b_{1}) : b_{1} ∈ B}.

Notation B.5. Logical quantifiers ∃x(P (x)) is used to indicate that there exists some x with P (x), while ∀x(P (x)) indicates that every x has P (x). These “quantifiers” are embellished in a number of ways. For instance

∃^{∞}x(P (x))

indicates that there are infinitely many different x’s with P, while

∀^{∞}x(P (x))

is used when all but finitely many x’s have P – that is to say, ∀^{∞}x(P (x)) is equivalent to

¬∃^{∞}¬x(P (x)).

Alternatively

∃x ∈ X(P (x))

is used to assert that there is some element x of the set X with P (x) whilst

∀x ∈ X(P (x))

asserts that no element of X fails to have P.

The categoricity quantifiers ∃^{∗} and ∀^{∗} are defined at 2.45.

Notation B.6. Unions of sets If A is a set of sets we let

⋃ A

be the union of all sets contained in A. If

x ↦ A_{x}

is an assignment of sets to elements of a space X, then for any Y ⊂ X

⋃_{x∈Y} A_{x}

indicates the union of the set of sets {A_{x} : x ∈ Y.}.

Of course one could iterate the union operation, and we have

⋃ ⋃ A

as the union of all sets contained in ⋃ A, and so on. The end result of iterating this process infinitely often and taking all sets arising this way is denoted TC(A). Thus TC(A), the transitive closure of A, is the collection of all sets appearing as an element somewhere in the hierarchy

A,
⋃ A,
⋃ ⋃ A,
⋃ ⋃ ⋃ A,
·
·
·

Alternatively one may define the transitive closure of A to be the smallest transitive set including A.

Thus if we have A = {{1, {2, 4}}, 6, {1, 3, 4}}, the set whose members are exactly {1, {2, 4}}, 6, {1, 3, 4}, then TC(A) would be

{0, 1, 2, 3, 4, 5, 6, {2, 4}, {1, 3, 4}, {1, {2, 4}}}.

The hereditarily countable sets, HC, are all sets A for which TC(A) is countable.

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192 BIBLIOGRAPHY

Index

≈, 4.5 ≈′, 4.5 ≈′′, 4.5 ≈_{n,m}^{x}, 6.22 ℵ_{0}, A.6f absolutely Δ ∼_{2}^{1}, 9.1 action by conjugation (of a Polish group on

itself) Hom^{+}([0, 1]), §4.2 Hom([0, 1]), §4.2 Hom([0, 1]^{2}), §4.3 M_{∞}, 0.3, 1.6, 2.12 S_{∞}, 3.11 U_{n}, 0.2 U_{∞}, 1.1, 2.12, 3.27

actions by specific classes of Polish groups abelian, §7.2, 7.23, 7.24 c_{0}, 2.66, 3.23, 3.34 c_{00}, 8.2, 8.3 CLI (admitting a complete left invariant

metric), 7.52 compact, 7.17 countable discrete, 2.23, 2.24, 2.64, 3.7,

3.8 density ideal, 3.26, 3.34 direct limits of locally compact, 7.45 Hom^{+}([0, 1]), 10.6 invariantly metrizable, 7.23, 7.24 l^{1}, 3.23, 3.25, 3.33 locally compact, 7.10, 7.44, 10.12 M_{∞}(ℝ), §6.3 monothetic, 3.28 nilpotent, 7.25, 7.26 products of locally compact, §7.4 S_{∞} (and its closed subgroups), 2.37, 2.39,

2.41, §6.1 solvable, 7.19, 7.27, 7.53 summable ideal, 3.26, 3.33 ℚ, 3.8, 7.22 (ℚ^{+}, ·), 3.10 U_{∞}, §5, 10.7 ℤ_{2}^{<ℕ}, 7.9 ℤ^{ℤ}, 7.27

analytic set, see Σ ∼_{1}^{1} atomic models, 7.21(ii)

B_{ϵ}(x), 4.2 B_{2}, 2.22 B_{∞}, 8.6 B^{+}, 7.32 B^{♯}, 7.32 Baer’s theorem on rank one TFA groups,

3.10f, §3.4.1 Baire measurable function, 2.48 Baire property, the, 2.45 Becker-Kechris theorem

on changing topologies, 7.38

on classification by countable models, 2.39,

6.19 on reducing E_{0}, 7.7

Becker’s theorem on CLI groups, §7.5 Birkhoff-Kakutani, 7.2 Borel equivalence relation, 2.21 Borel function, 2.1 Borel G-space, 2.5 Borel group, 2.7 Borel reduction, 2.21 Borel set, 2.1 Burgess’ theorem on existence of orbit in-

verses, 2.27

c_{0}, 2.6, 3.23 c_{00}, 8.3 ≅^{c} (equivalence relation of homeomorphism

on compact metric spaces) definition, 4.19 reducible to a group action, §4.4

≅^{i} (equivalence relation of isometry on com-

pact metric spaces) definition, 4.19 smoothness, §4.4

characteristic function, B.3 CLI groups, 7.47, §7.5 classification by countable models, see clas-

sification by countable structures classification by countable structures, 2.38 comeager, 2.45 compatible metric, 2.1 coset equivalence relation, 2.12 countable, A.6

AΔB, 2.2, 3.26 d(V), the diameter of V, B.4 d(x, C), B.4 density ideal, 3.26 dichotomy theorems, §3.4.3, 3.30

E_{G}^{X}, 2.5 ∃, 2.35 ∃^{∗}, 2.45 E_{0}, 2.22, 2.29 E_{1}, 8.1 ∅, A.2 Effros Borel structure, 2.4 Effros lemma on G_{δ} orbits, 7.12 Effros theorem on locally compact group ac-

tions, 7.10 embedding (G-embedding for a group G),

6.39

∀, 2.35 ∀^{∗}, 2.45 F_{2}, F_{k}, 2.22 F_{σ}, 2.1 F_{σδ}, 2.1 F(X), 2.4 f_{x}^{O}, 7.30 faithful Borel reduction, 6.38 Friedman-Stanley theorem on linear orders,

4.10 Friedman-Stanley theorem on the cosets of

the density ideal in P(ℕ), §3.4.3, 9.9

γ(x), 6.23 γ^{∗}(x), 6.3 g · A, 2.5 G_{δ}, 2.1 G_{δσ}, 2.1 Ĝ, 4.15 G_{x}, the stabilizer of x, 2.5 GE group, 6.36 generically F -ergodic, 3.6 generically turbulent, 3.20 Glimm-Effros property, 7.13 (strong), 7.46

(weak) Gromov-Hausdorff topology, §4.4

HC, 0.11, 2.41f, 6.5f, B.6 Harrington-Kechris-Louveau, 3.30 Hausdorff theorem on open images of Polish

spaces, 7.5 Hilbert space, 2.9 Hom^{+}([0, 1]), 4.1 Hom([0, 1]), 4.1 Hom([0, 1]^{2}), 4.1 Hom(K), 4.1 homogeneous compact metric space, 7.20

⇒, 2.35 I_{2}, 3.26 I_{d}, 3.26 id(X), 2.22 ideals, 3.26 interior of a set, B.4 index (of a subgroup), 2.62 invariant, 2.5 invariant metric, 7.1, 7.23 irreducible (representation), 5.4

Jankov-von Neumann uniformization, 2.54

Kuratowski-Ulam, 2.46 Kuratowski-Mylcielski, 2.58 Kechris-Louveau theorem on E_{1}, 8.2

l^{1}, 2.8 L^{2}, 2.9 ≤_{aΔ_{2}^{1}}, 9.1 ≤_{α}, 6.48 ≤_{B}, 2.21 ≰_{B}, 2.21 ≰_{c}, 2.21 ≤_{c}, 2.21 ≤_{F B}, 6.38 ≤_{L(ℝ)}, 9.17 ⊑_{B}, 2.21 ⊑_{c}, 2.21 L(G, d), 7.28 L(ℝ)-cardinality, | · |_{L(ℝ)}, 9.10 left invariant metric, 7.1 limit ordinal, A.1 local orbit, 3.15 logic action, 2.37

M_{∞}, 2.9 M_{∞}(ℝ), 6.43 Martin-Moschovakis-Steel, 9.12 Mod(L), 2.33 Mod(L, κ), 9.17 ⊧, 2.35 meager, 2.45

¬, 2.35 ℕ = {0, 1, 2, ...}, 2.2 ℕ^{ℕ}, 2.2 ℕ^{n}, 2.2 ℕ^{<ℕ}, 2.2

O(x, U, V), 3.15 ω_{1}, A.6 \overline{A} (the closure of A), 2.31(ii), B.4 1, A.2 ordinal, A.1

P, 3.26, B.1 P_{ℵ_{0}}(A) (the countable subsets of A), 2.32 Π ∼_{α}^{0}, 2.1, §A φ_{0}(x, U, V), 3.15 φ_{α}(x, V_{n}), 6.1 φ_{α}(x, U, V), 6.21 φ_{x}^{∗}, 6.4 φ_{x}, 6.23 π_{O}, 7.31 perfect set, 2.57 Pettis’ lemma, 2.56 Polish group, 2.5 Polish space, 2.1 Polishable group, 2.7 potential Borel complexity, 2.66, 6.12 products of locally compact, §7.4 properly generically ergodic, 3.1 property of Baire, see Baire property

reduction to F_{2}, 2.23, 2.32, 3.14, 7.44 of F_{2}, 2.31, 2.64 to P_{ℵ_{0}}(ℝ), 2.32, 0.3, 0.4 to P_{ℵ_{0}}(ℂ), 0.3 to id(ℕ), 0.1 to id(ℝ), 0.2, 2.26, 2.27, 7.9, 7.10, of id(ℝ), 2.27, 2.40, to isomorphism on countable models (or

isomorphism on countable structures), preface, §1.1, 2.38, 2.39, 2.40, 2.40, 3.19, 3.21, §3.4.3, 6.19, 9.15

to ≅|_{Mod(L)}, see reduction to isomorphism

on countable models to a Polish group action, 8.2, 10.9, 10.10,

10.11 representation, 5.1, 10.13 ∼_{r}, 5.1, ∼_{r} not classifiable by countable structures,

5.9 right invariant metric, 7.1

S_{∞}, 2.9 S_{κ}, 9.17 ⊧ (satisfaction relation), 2.35 X \ O, 2.3 Σ ∼_{1}^{1}, 2.52 Σ ∼_{α}^{0}, 2.1, §A [x]_{E}, [x]_{G}, 2.21 [g]_{n,m}^{x}, 6.22 [A]_{G} =_{df} G · A (the saturation of A), 2.5 Scott analysis, §3.4.2, §6.1 Scott sentence, §3.4.2 shift action, 2.24 smooth, 2.26 Solecki’s theorem on invariantly metrizable

groups, 7.23 spectral theorem, 10.7f stabilizer, 2.5 standard Borel space, 2.1 strong Glimm-Effros property, 7.13 successor ordinal, A.1 summable ideal, 3.26 super dense, 3.24

2 = {0, 1}, 2.1, A.2 2^{G} (functions from G to {0, 1}), 2.24 2^{ℕ}, 2.1 2^{<ℕ}, 2.1 2^{n}, 2.1 𝕋, 7.22 τ(φ_{x}), 6.31 TC (transitive closure), 2.41f, 6.5f, B.6 torsion free abelian groups, rank one, 3.10 torsion free abelian groups, finite rank, §3.4.1 transfinite induction, A.5f transitive closure, 2.41f, 6.5f, B.6 turbulent, 3.13

U_{∞}, 2.9

U(φ_{α}(x, U_{n}, V_{m}), n, m), 6.31

UBMC (universally Baire measurable in the

codes), 3.12 uniformization

by a universally measurable function, 2.54 by Baire measurable function, 2.54 universally Baire measurable function, 2.48, unitary group, 2.9

∨, 2.35

Vaught transforms B^{∗V} , 2.51, 6.7-6.9 B^{ΔV} , 2.51, 6.7-6.9

Vitali equivalence relation, 7.22

W^{−1}, 2.13 ∧, 2.35 weak Glimm-Effros property, 7.46

X/G, 2.21 x(n, ·), 2.22 X(φ_{x}), 6.31 X̂(O), 7.34

ℤ_{2}, 2.6, 3.26 ℤ_{2}^{ℕ}, 3.26 0, A.2