A converse to Dye’s theorem^{∗}

Greg Hjorth^{†}

Abstract

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of 𝔽_{2} on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ≤_{B} ordering.

1 Introduction

We show a converse to a consequence of the final strengthening of Dye’s theorem proved by Ornstein and Weiss.

Definition An equivalence relation E is said to be standard if it is defined on a standard Borel probability space (X, B, μ), it is Borel as a subset of X × X, and all its equivalence classes are countable. It is said to be measure preserving if for any measurable A, B ⊂ X and for any measurable bijection φ : A → B included in the graph of E (xEφ(x) all x) we have μ(A) = μ(B). It is ergodic if any E-invariant set is either null or conull.

Two such equivalence relations, E on (X, B, μ), F on (Y, C, ν), are said to be orbit equivalent if there is a measure preserving bijection

ψ : X → Y

such that almost everywhere we have

x_{1}Ex_{2} ⇔ ψ(x_{1})F ψ(x_{2}).

An equivalence relation E on a standard Borel probability space (X, B, μ) is said to be induced by a countable group G if there is a Borel, measure preserving, ergodic, and almost everywhere free action of G on X such that E equals the corresponding equivalence relation E_{G}. Any such E_{G} will necessarily be standard, measure preserving, and ergodic.

Theorem 1.1 (Dye; see [5], [6].) Any two ergodic, standard, measure preserving equivalence relations induced by ℤ are orbit equivalent.

Theorem 1.2 (Ornstein, Weiss; see [14], [3].) If G is a countably infinite amenable group, then any two ergodic, standard, measure preserving equivalence relations induced by G are orbit equivalent.

Theorem 1.3 (Connes, Weiss; Schmidt; see [4], [17].) If G is countable, non-amenable and without property T, then there are at least two orbit inequivalent ergodic, standard, measure preserving equivalence relations induced by actions of G.

Below:

∗Key words and phrases: Ergodic theory, treeable equivalence relations, non-amenable groups, property T groups, free groups, Borel reducibility. AMS Subject Classification: 03E15, 28D15, 37A15.

†Partially supported by NSF grant DMS 01-40503.

G. Hjorth 2

Theorem 1.4 If G is countably infinite with property T, then there are continuum many orbit inequivalent ergodic, standard, measure preserving equivalence relations induced by actions of G.

Corollary 1.5 A countable group is amenable if and only if it induces only one orbit equivalence relation considered up to orbit equivalence.

The argument, which answers a question raised by Schmidt at 3.10 of [18], was inspired by certain constructions from [15].

We also consider another structural consequence of Dye’s work, who in the course of the proof of his theorem provided a normal form for the orbit equivalence relations induced by groups such as ℤ. Any measurable equivalence relation induced by ℤ can be represented as an orbit equivalence relation arising by a kind of inverse limit of actions of ℤ on finite spaces. While it is unreasonable to hope that the specifics of Dye’s construction, with ℤ acting by the odometer map as in [5], could provide a canonical model for non-amenable equivalence relations, it does seem to have been open whether arbitrary measure preserving standard ergodic E may allow themselves to be presented as arising from this kind of inverse limit of actions on finite spaces.

We formalize the notion of modular in section 3 to capture this idea, and go onto show that in general 𝔽_{2} can induce measurable equivalence relations which are not modular in this sense. It turns that the mixing properties of the Bernoulli shift of 𝔽_{2} on 2^{𝔽_{2}}, in direct contrast to the Bernoulli shifts of amenable groups, are somehow “remembered” at the level of orbit equivalence, and do not allow themselves to be modeled by a modular equivalence relation.

This argument turns out to give information in the context of Borel reducibility. In answer to a question of Jackson, Kechris, and Louveau:

Theorem 1.6 There is a countable treeable Borel equivalence relation which is not hyperfinite and does not Borel reduce every other countable treeable Borel equivalence relation.

Here we say that E Borel reduces F if there is a Borel function between their respective fields with x_{1}Ex_{2} if and only if θ(x_{1})F θ(x_{2}); roughly speaking, the quotient space from E Borel injects into the quotient space from F. We say that an equivalence relation is hyperfinite if it can be written as the increasing union of Borel equivalence relations with finite classes; equivalently, see [13], that it is induced by a Borel action of ℤ. We say that E is treeable if there is an acyclic Borel graph on its field whose connected components form the E-equivalence classes; equivalently, there is a collection of partial Borel bijections which generate E and allow no non-trivial loops.

Acknowledgments: I am grateful to John Clemens, Damien Gaboriau, Alexander Kechris, and Simon Thomas for helpful remarks about earlier drafts of this paper and pointing out various errors. With kind permission many of their comments have been included in the revised version. I am also grateful to Sergey Bezuglyi for his help with the references.

2 Orbit equivalence relations induced by Kazhdan groups

Notation We generally write U(H), or U_{∞}(H) when we know H to be infinite dimensional, for the group of unitary transformations of a Hilbert space H.

Definition Let

π : G → U(H)
g ↦ π_{g}

be a unitary representation of a group G. This representation is said to have almost invariant unit vectors if for all ϵ > 0 and F ⊂ G finite there is some ζ ∈ H with ||h|| = 1 and ||ζ − π_{g}(ζ)|| < ϵ all g ∈ F.

Definition A discrete group G is said to be Kazhdan or to have property T if any unitary representation π : G → U(H) with almost invariant unit vectors has an outright invariant unit vector – that is to say, some ζ ∈ H with ||ζ|| = 1 and

π_{g}(ζ) = ζ

all g ∈ G.

Possible references for the subject of Kazhdan groups are given by [9] and [22]. We mention two apparent strengthenings of the definition which in fact turn out to be equivalent. In both cases the proofs are routine and can be left as exercises for the reader.

Proposition 2.1 G has property T if and only if there is some finite F ⊂ G and ϵ > 0 such that whenever π : G → U(H) is a unitary representation with some ζ ∈ H having ||π_{g}(ζ) − ζ|| < ϵ all g ∈ F then π has an invariant unit vector.

Proposition 2.2 G has property T if and only if for all δ > 0 there is some finite F ⊂ G and ϵ > 0 such that whenever π : G → U(H) is a unitary representation with some ζ ∈ H having ||π_{g}(ζ) − ζ|| < ϵ all g ∈ F then π has an invariant unit vector η with ||η − ζ|| < δ.

One natural example of a Kazhdan group is the collection of three by three integer coefficient matrices with determinant one. More generally, at every n ≥ 3 the group SL_{n}(ℤ) is Kazhdan.

In say chapter 7 of [22] one can find an extended discussion of theorems to the effect that certain kinds of discrete subgroups of certain kinds of Lie groups will, under the appropriate conditions, be Kazhdan. As with Kazhdan’s original proof for SL_{3}(ℤ), the proofs are analytical in flavor, turning on the topological properties of the ambient Lie group. More recently Andre Zuk in [21] has obtained purely combinatorial proofs that certain discrete groups have property T. Indeed he even shows that in some suitably statistical sense most finitely generated groups in an indicated class are Kazhdan.

In this section we prove that all countably infinite Kazhdan groups have continuum many free, ergodic, measure preserving actions on standard Borel spaces up to orbit equivalence. This theorem was previously known from [8] for certain special classes of property T groups; for instance it was known for SL_{3}(ℤ). The first proof that there is at least some countable group with continuum many actions can be found in [2], which built on work by McDuff in the theory of operator algebras. Our proof is more elementary than these previous arguments.

Definition Let E be a standard, measure preserving equivalence relation on (X, B, μ) and let G be a countable group.

We then set C(E, G) to be the collection of all measurable

α : E → G

such that almost everywhere (i) if y ∈ [x]_{E} then α(y, x) = 1_{G} if and only if y = x;

(ii) if y, z ∈ [x]_{E}, then α(z, y)α(y, x) = α(z, x);
(iii) for all g ∈ G there exists y ∈ [x]_{E} with α(y, x) = g.

We identify α_{1}, α_{2} ∈ C(E, G) if they agree almost everywhere.

Remark We can think of C(E, G) as the space of possible ways to arrange a free action and measurable action of G on X with E_{G} = E.

In the case that Γ induces E, we can identify C(E, G) with the cocycles from X × Γ → G which are appropriately “one-to-one” and “onto”.

Definition From now on fix an enumeration (g_{n})_{n∈ℕ} of the group G. For E as above, and α, β ∈ C(E, G) we let

d_{E}(α, β) = ∑_{n∈ℕ}2^{−n}μ({x : ∃y(α(y, x) = g_{n}, β(y, x) ≠ g_{n})} ⋃{x : ∃y(β(y, x) = g_{n}, α(y, x) ≠ g_{n})}).

C(E, G) equipped with this metric becomes a separable, complete metric space. We will not need the completeness of the metric, but the separability plays a starring role.

Fact 2.3 For E and G as above, C(E, G) has a countable dense subset.

Proof Let us consider C_{0}(E, G), the set of measurable

α : E → G

with |{y : α(y, x) = g}| < 2 for all g ∈ G, a.e. x ∈ X. Since C(E, G) is included in C_{0}(E, G) and the metric for the first extends to a metric for the second, it will suffice to find a countable dense subset of C_{0}(E, G).

For this purpose, let (f_{n})_{n∈ℕ} be a sequence of measurable injections with (f_{n}(x))_{n∈ℕ} enumerating [x]_{E} almost everywhere. Let B_{0} ⊂ B be a countable Boolean subalgebra which is dense with respect to the measure algebra, in the sense that for all B ∈ B, ϵ > 0, there is some B_{0} ∈ B_{0} with μ(BΔB_{0}) < ϵ. Then consider the collection of all α ∈ C_{0}(E, G) satisfying:

(i) for each g, the set of n with μ({x : α(f_{n}(x), x) = g}) ≠ 0 is finite;

(ii) for each g, n, the set {x : α(f_{n}(x), x) = g} is in B_{0};
(iii) for all sufficiently large n, {x : α(f_{n}(x), x) = g_{n}} = X.

The countability and density of this collection are routinely verified. □

Note that the metric d_{E} depends on the choice of the enumeration (g_{n})_{n∈ℕ} of G. In the observations which follow we will want to consider a single countable group G in relation to various choices of E. We will think of a countable group as coming with some fixed choice of an enumeration (g_{n})_{n∈ℕ}, and in each case use d_{E} to refer to the metric which arises on C(E, G) for that predetermined choice; we will not specifically mention the enumeration.

Definition (Compare [3].) For E a standard, measure preserving equivalence relation on (X, B, μ), let p_{1}, p_{2} : E → X be the projections onto the first and second coordinates. We then define a measure m on E by

m(B) = ∫_{X} |p_{1}^{−1}[{x}] ∩ B|dμ(x).

Since E is measure preserving, we could as well have used the projection p_{2} in place of p_{1} and obtained the same measure.

We then let U(E) be the collection of all unitary operators on the Hilbert space L^{2}(E, m) of square integrable f : E → ℂ. Given α, β ∈ C(E, G) we define a unitary representation

π^{α,β} : G → U(E),
g ↦ π_{g}^{α,β}

by

(π_{g}^{α,β}(f))(x′, y′) = f(x, y)

where x′, y′ are defined by the specification that

α(x′, x) = g,
β(y′, y) = g.

In other words, if we use α and β in the obvious way to obtain actions

a_{α} : G × X → X,
a_{β} : G × X → X,

and we take the induced measure preserving transformation

a_{α} × a_{β} : G × E → E

given by

(a_{α} × a_{β})(g, (x, y)) = (a_{α}(g, x), a_{β}(g, y)),

then in the usual manner we produce a representation of G in U(E).

Notation In what follows, χ_{Δ} : E → {0, 1} is the characteristic function of the diagonal; so that χ_{Δ}(x, y) = 1 if x = y and 0 if x ≠ y.

Fact 2.4 Let G be a countably infinite group. For all ϵ > 0 and finite F ⊂ G, there exists a δ > 0 such that for all standard, measure preserving E, all α, β ∈ C(E, G), if d_{E}(α, β) < δ, then

∀g ∈ F (||π_{g}^{α,β}(χ_{Δ}) − χ_{Δ}|| < ϵ),

where || · || refers to the Hilbert space norm on L^{2}(E, m).

(□)

Lemma 2.5 Let G be a countably infinite group with property T.

Then there is a δ > 0 such that for all ergodic, standard, measure preserving E on (X, B, μ) and all α, β ∈ C(E, G) with

d_{E}(α, β) < δ,

there is a measurable bijection φ : X → X, φ ⊂ E, such that for the induced actions

a_{α} : G × X → X,
a_{β} : G × X → X,

we have

a_{α}(g, x) = φ^{−1}(a_{β}(g, φ(x)))

almost everywhere.

Proof Since G has T, we may find ϵ > 0 and finite F ⊂ G such that whenever π : G → U_{∞}(H) is a unitary representation and η ∈ H is a unit vector with

∀g ∈ F (||π_{g}(η) − η|| < ϵ),

then there is a unit vector η_{0} ∈ H which is G-invariant and has

||η − η_{0}|| < 10^{−2}.

For this F ⊂ G and ϵ > 0 we choose δ > 0 as 2.4.

We then consider E as above, α, β ∈ C(E, G) with d_{E}(α, β) < δ. By the assumption on δ, ϵ, F, we may find G-invariant f ∈ L^{2}(E, m) with

||χ_{Δ} − f|| < 10^{−2}.

Consider then the set

A_{f} = \{x ∈ X|∃!y ∈ [x]_{E}(|f(x, y) − 1| < \frac{1}{4} )\}.

Since f is close to χ_{Δ}, A_{f} is non-null. But then by ergodicity of action a_{α} : G × X → X we must have A_{f} co-null.

G-invariance of f amounts to the assertion that for almost all (x, y) ∈ E and all g ∈ G

f(x, y) = f(a_{α}(g, x), a_{β}(g, x)).

Thus if we define φ : A_{f} → X by φ(x) = y if and only if |f(x, y) − 1| < \frac{1}{4} then this equation implies

φ(x) = y ⇔ φ(a_{α}(g, x)) = a_{β}(g, y),

which in turn gives

a_{α}(g, x) = φ^{−1}(a_{β}(g, φ(x)))

almost everywhere. □

Corollary 2.6 If E is an ergodic, standard, measure preserving, equivalence relation on (X, B, μ) and G is a property T group, then up to isomorphism^{1} there are at most ℵ_{0} many free actions of G by measurable transformations on X which induce E.

Proof We may identify such actions with elements of C(E, G). The last lemma asserts that the equivalence relation of isomorphism of action is open in C(E, G), and so the corollary follows from the separability of this space. □

Corollary 2.7 If G is a countably infinite property T group, then it induces 2^{ℵ_{0}} many orbit inequivalent ergodic, standard, measure preserving, equivalence relations.

Proof Since the group G is not abelian-by-finite, and hence not type I (see [20]), we may find

π_{γ} : G → U_{∞}(H_{γ}),

γ ∈ 2^{ℵ_{0}}, non-conjugate infinite dimensional irreducible unitary representations of G. Following [22] 5.2.13, we may find corresponding ergodic actions of G on standard Borel probability spaces (X_{γ}, B_{γ}, μ_{γ}), with each induced representation σ_{γ} : G → U_{∞}(L^{2}(X_{γ}, μ_{γ})) having H_{γ} as a direct summand. The product of an ergodic action of an infinite group with a mixing action is still ergodic, so after possibly taking the product of this action with the Bernoulli shift, of G acting on {0, 1}^{G}, we may assume that each action is free.

Then each induced representation σ_{γ} : G → U_{∞}(L^{2}(X_{γ}, μ_{γ})) has only countably many irreducible representations as direct summands, and is therefore only isomorphic as a unitary representation to countably many other σ_{κ}’s; in particular the original measure preserving action from which it derives is only isomorphic to countably many others. Thus by 2.6 each such E_{σ_{γ}} is orbit equivalent to at most countably many other such E_{σ_{α}}’s. □

3 Non-modular equivalence relations induced by 𝔽_{2}

In this section we formulate the notion of a modular equivalence relation, and show that in general the free actions of 𝔽_{2} may induce non modular actions.

The argument gives information in the context of Borel reducibility, and thus in answer to question 6.4(A) from [13], we obtain in the following section a treeable countable Borel equivalence relation which is neither universal treeable nor hyperfinite.

Very recently Damien Gaboriau and Sorin Popa have shown that any non-abelian free group gives rise to continuum many non-orbit equivalent free actions. As far as can be determined, their argument does not appear to provide insight at the level of ≤_{B}-reducibility, and it remains open whether 𝔽_{2}, or any 𝔽_{n} (n = 2, 3, ..., ℵ_{0}) has more than two non-hyperfinite free Borel actions up to Borel reducibility.

There is a fair amount in way of prefatory lemmas and definitions. We need to distinguish the highly mixing actions of say the Bernoulli shift from those actions which have generating sets for the Borel algebra with finite orbit. We make that distinction by assigning to each measurable set in 2^{𝔽_{2}} =_{df} {0, 1}^{𝔽_{2}} a center, cradled between those elements of 𝔽_{2} which are most important for its definition. The various prefatory lemmas are required for that definition and to formulize the manner in which this center is rapidly mixed by the equivalence relation.

We think of 𝔽_{2} = ⟨a, b⟩ as having a and b as generators.

Definition A weight for 𝔽_{2} is a function

w : 𝔽_{2} → ℝ

such that w(σ) ≥ 0 all σ ∈ 𝔽_{2}, w is non-zero at some point, and w ∈ ℓ^{1}(𝔽_{2}). An element σ_{0} is a center for some such weight w if each

∑{w(τxσ_{0}) : τ ∈ 𝔽_{2}, τx is a reduced word},

1Here we say that two actions of G are isomorphic if they are simultaneously conjugate: That is to say, a, b : G × X → X are isomorphic actions if there is measure preserving φ : X ≅ X with φ(a(g, φ^{−1}(x))) = b(g, x) almost everywhere.

as x ranges over {a, a^{−1}, b, b^{−1}}, is less than

\frac{1}{2} ∑\{w(τ) : τ ∈ 𝔽_{2}\}.

σ_{0} is a weak center if each such

∑\{w(τxσ_{0}) : τx is a reduced word\} < \frac{2}{3} ∑\{w(τ) : τ ∈ 𝔽_{2}\}.

Lemma 3.1 Every weight has at least one center.

Proof We begin with some e ∈ 𝔽_{2}, and if that is a center we are content.

Otherwise there will be some x_{0} ∈ {a, a^{−1}, b, b^{−1}} with most of the weight lying on reduced words of the form τx_{0} and so we shift our attention to x_{0}. And again if x_{0} is a center we are done, and otherwise we pass onto some x_{1} with most of the weight lying on words of the form τx_{1}x_{0}. Since the majority of the weight lies within some definite distance from the identity, this process must terminate after considering finitely many x_{i}’s. □

Notation We let 𝔽_{2} act on ℓ^{1}(𝔽_{2}) by right multiplication:

τ · f(σ) = f(στ).

Lemma 3.2 σ_{0} is a center for a weight w if and only if σ_{0}τ^{−1} is a center for τ · w; σ_{0} is a weak center for a weight w if and only if σ_{0}τ^{−1} is a weak center for τ · w.

Proof Immediate from the structure of the definitions. □

Note then that the centers of a weight and the weak centers of a weight form a finite subset of 𝔽_{2} which is convex when viewed as being included in the Cayley graph. In fact they are not just convex but linear; this is the key combinatorial fact.

Lemma 3.3 If w is a weight, then its collection of weak centers is linearly ordered; that is to say, there are x_{0}, x_{1}, ..., x_{n}, with x_{0} ∈ 𝔽_{2}, each x_{i+1} ∈ {a, a^{−1}, b, b^{−1}}, such that each x_{i}x_{i−1}...x_{0} forms a reduced word and

{x_{i}x_{i−1}...x_{0} : i ≤ n}

enumerates the weak centers.

Proof Assume instead that there is some σ_{0} ∈ 𝔽_{2} such that aσ_{0}, a^{−1}σ_{0}, bσ_{0} are all weak centers. (Without any real loss of generality we can assume that this is the case; the other possibilities are entirely similar.)

Then for c ∈ {a, a^{−1}} we have for x = c^{−1} that

∑\{w(τxcσ_{0}) : τx is a reduced word\} < \frac{2}{3} ||w||_{ℓ^{1}}

∴ ∑\{w(τcσ_{0}) : τc is a reduced word\} > \frac{1}{3} ||w||_{ℓ^{1}}.

For y = b^{−1} we have

{τ_{0}ybσ_{0} : τ_{0}y reduced} ⊃ {τ_{1}aσ_{0} : τ_{1}a reduced} ∪ {τ_{2}a^{−1}σ_{0} : τ_{2}a^{−1} reduced}

∴ ∑\{w(τybσ_{0}) : τy is a reduced word\} > \frac{2}{3} ||w||_{ℓ^{1}},

with a slur on bσ_{0}’s claim of weak centerhood. □

Notation We let 2^{𝔽_{2}} =_{df} {0, 1}^{𝔽_{2}} be the collection of functions from 𝔽_{2} to {0, 1}; we equip this with the product measure and let 𝔽_{2} act by multiplication on the right: For f ∈ 2^{𝔽_{2}}, σ, τ ∈ 𝔽_{2}

τ · f(σ) = f(στ).

This in turn induces an action on L^{2}(2^{𝔽_{2}}) in the usual way: Given

ψ : 2^{𝔽_{2}} → ℂ

we define τ · ψ by the formulation

τ · ψ(f) = ψ(τ^{−1} · f).

For S ⊂ 𝔽_{2} finite we define a corresponding function

ψ_{S} : 2^{𝔽_{2}} → {−1, 1}
f ↦ (−1)^{|{σ∈S:f(σ)=1}|}.

Thus if f assumes the value 1 on an odd number of elements of S then ψ_{S}(f) = −1; otherwise it takes the value 1.

It is well known that {ψ_{S} : S ⊂ 𝔽_{2} is finite} forms an orthonormal basis. One can for instance determine by hand that they are orthogonal and then by Stone-Weierstrass check that their linear combinations are dense, and hence that they span.

Note that ψ_{∅} : f ↦ 1 is the constant function with value 1. ψ ∈ L^{2}(2^{𝔽_{2}}) will be orthogonal to the constant functions if and only if it is in the subspace given by the closed span of {ψ_{S} : S ≠ ∅}.

Notation Let F(𝔽_{2}) = {S ⊂ 𝔽_{2} : S is finite}.

F(𝔽_{2}) is a countable set, and we can thus form ℓ^{2}(F(𝔽_{2})) in the usual way. This Hilbert space is of course naturally isomorphic to L^{2}(2^{𝔽_{2}}) and we introduce some notation to describe that isomorphism.

Notation Let

Δ : L^{2}(2^{𝔽_{2}}) → ℓ^{2}(F(𝔽_{2}))

be the linear isometry induced by the association

ψ_{S} ↦ δ_{S},

where δ_{S}(T) = 1 if T = S, = 0 if T ≠ S.

From here we wish to convert the elements of ℓ^{2}(F(𝔽_{2})) into weights. Of course now there is no question of a linear isomorphism; instead we ask only for a process of conversion which is Lipschitz on the unit sphere. We begin by mapping ℓ^{2}(F(𝔽_{2})) into the positive part of ℓ^{1}(F(𝔽_{2})).

Notation We define

E : ℓ^{2}(F(𝔽_{2})) → ℓ^{1}(F(𝔽_{2}))

by

E(ϕ)(s) = |ϕ(s)|^{2}.

Note that ||E(ϕ)||_{ℓ^{1}} = ||ϕ||_{L^{2}}^{2}.

Lemma 3.4 For ϕ_{1}, ϕ_{2} ∈ ℓ^{2}(F(𝔽_{2}))

||E(ϕ_{1}) − E(ϕ_{2})||_{ℓ^{1}} ≤ (||ϕ_{1}||_{ℓ^{2}} + ||ϕ_{2}||_{ℓ^{2}})||ϕ_{1} − ϕ_{2}||_{ℓ^{2}}.

Proof

||E(ϕ_{1}) − E(ϕ_{2})||_{ℓ^{1}} = ∑_{s∈F(𝔽_{2})}|E(ϕ_{1})(s) − E(ϕ_{2})(s)|

= ∑_{s∈F(𝔽_{2})}|ϕ_{1}(s)^{2} − ϕ_{2}(s)^{2}| = ∑_{s∈F(𝔽_{2})}|ϕ_{1}(s)[ϕ_{1}(s) − ϕ_{2}(s)] + ϕ_{2}(s)[ϕ_{1}(s) − ϕ_{2}(s)]|

≤ ∑_{s∈F(𝔽_{2})}|ϕ_{1}(s)| · |ϕ_{1}(s) − ϕ_{2}(s)| + ∑_{s∈F(𝔽_{2})}|ϕ_{2}(s)| · |ϕ_{1}(s) − ϕ_{2}(s)|

= ⟨|ϕ_{1}|, |ϕ_{1} − ϕ_{2}|⟩ + ⟨|ϕ_{2}|, |ϕ_{1} − ϕ_{2}|⟩ ≤ ||ϕ_{1}||_{ℓ^{2}}||ϕ_{1} − ϕ_{2}||_{ℓ^{2}} + ||ϕ_{2}||_{ℓ^{2}}||ϕ_{1} − ϕ_{2}||_{ℓ^{2}},

by Cauchy-Schwarz. □

Notation We define

Σ : ℓ^{1}(F(𝔽_{2})) → ℓ^{1}(𝔽_{2})

by

Σ(ϕ)(σ) = ∑_{S∈F(𝔽_{2}),σ∈S} |S|^{−1}ϕ(S).

Σ is a linear contraction, and for ϕ positive, say in the range of E, of unit length, and with ϕ(∅) = 0, we have that Σ(ϕ) is a weight with ||Σ(ϕ)||_{ℓ^{1}} = ||ϕ||_{ℓ^{1}}.

We then let

π = ΣEΔ : L^{2}(2^{𝔽_{2}}) → ℓ^{1}(𝔽_{2}),
φ ↦ Σ(E(Δ(φ))).

Notation For S ⊂ 𝔽_{2} finite and τ ∈ 𝔽_{2} we let

τ · S = {στ^{−1} : σ ∈ S};

thus

σ ∈ S ⇔ στ^{−1} ∈ τ · S,
∴ στ ∈ S ⇔ σ ∈ τ · S.

Then for f ∈ ℓ^{2}(F(𝔽_{2})) ∪ ℓ^{1}(F(𝔽_{2})) and τ ∈ 𝔽_{2} we define τ · f ∈ ℓ^{2}(F(𝔽_{2})) ∪ ℓ^{1}(F(𝔽_{2})) by

τ · f(S) = f(τ^{−1} · S);

thus if f = δ_{S_{0}} then

τ · f(S) = 1 iff τ^{−1} · S = S_{0}
iff S = τ · S_{0};

and thus τ · δ_{S_{0}} = δ_{τ·S_{0}}.

Lemma 3.5 (o) For φ ∈ L^{2}(2^{𝔽_{2}}) orthogonal to the constant functions and of norm one, ||π(φ)||_{ℓ^{1}} = 1. (i) ||π(φ_{1}) − π(φ_{2})||_{ℓ^{1}} ≤ (||φ_{1}||_{L^{2}} + ||φ_{2}||_{L^{2}})||φ_{1} − φ_{2}||_{L^{2}}. (ii) π is an 𝔽_{2}-map, in the sense that for σ ∈ 𝔽_{2} and φ ∈ L^{2}(2^{𝔽_{2}})

π(σ · φ) = σ · π(φ).

(iii) For φ not a.e. constant, π(φ) is a weight.

(iv) For such φ and σ ∈ 𝔽_{2}, σ_{0} is a center of π(φ) if and only if σ_{0}σ^{−1} is a center of π(σ · φ); σ_{0} is a weak center of π(φ) if and only if σ_{0}σ^{−1} is a weak center of π(σ · φ).

(v) For φ_{1}, φ_{2} unit vectors in L^{2}(2^{𝔽_{2}}) orthogonal to the constant functions with ||φ_{1} − φ_{2}||_{L^{2}} < \frac{1}{12} , and σ_{0} a center of π(φ_{1}), we will necessarily have σ_{0} a weak center of π(φ_{2}).

Proof (o) and (i) follow from the various facts recorded above regarding E, Δ, and Σ.

(ii): We first observe: Claim: Δ : L^{2}(2^{𝔽_{2}}) → ℓ^{2}(F(𝔽_{2})) is an 𝔽_{2}-map. Proof of claim: It suffices to check this for the basis elements, of the form ψ_{S}. So let τ ∈ 𝔽_{2}, S ⊂ 𝔽_{2} finite, f ∈ 2^{𝔽_{2}}.

τ · ψ_{S}(f) = ψ_{S}(τ^{−1} · f) = (−1)^{|{σ∈S:τ^{−1}·f(σ)=1}|}
= (−1)^{|{σ∈S:f(στ^{−1})=1}|} = (−1)^{|{στ∈S:f(σ)=1}|} = (−1)^{|{σ∈τ·S:f(σ)=1}|}
= ψ_{τ·S}(f).

Thus Δ(τ · ψ_{S}) = Δ(ψ_{τ·S}) = δ_{τ·S} = τ · δ_{S} = τ · Δ(ψ_{S}). (Claim□) It is immediate that E is an 𝔽_{2}-map, and as for Σ we have

(Σ(τ · f))(σ) = ∑_{σ∈S} |S|^{−1}τ · f(S) = ∑_{σ∈S} |S|^{−1} · f(τ^{−1} · S)

= ∑_{σ∈τ·S} |S|^{−1}f(S) = ∑_{στ∈S} |S|^{−1}f(S) = (Σ(f))(στ) = (τ · Σ(f))(σ).

(iii) This is obvious from the definitions and 3.4.

(iv) We have for any x ∈ {a, a^{−1}, b, b^{−1}} that

∑{π(σ · φ)(τxσ_{0}σ^{−1}) : τx is a reduced word} = ∑{σ · π(φ)(τxσ_{0}σ^{−1}) : τx is a reduced word}

by (ii) above,

= ∑{π(φ)(τxσ_{0}) : τx is a reduced word},

by the definition of the action.

(v) If ||φ_{1}||_{L^{2}}, ||φ_{2}||_{L^{2}} = 1, φ_{1}, φ_{2} ⊥ ℂ1, and

||φ_{1} − φ_{2}||_{L^{2}} < \frac{1}{12}

then using (0), (i) we have

||π(φ_{1}) − π(φ_{2})||_{ℓ^{1}} < \frac{1}{6} min\{||π(φ_{1})||_{ℓ^{1}}, ||π(φ_{2})||_{ℓ^{1}}\} = \frac{1}{6} .

Thus for each σ_{0} ∈ 𝔽_{2}, x ∈ {a, a^{−1}, b, b^{−1}}

| ∑{π(φ_{1})(τxσ_{0}) : τx reduced } − ∑{π(φ_{2})(τxσ_{0}) : τx reduced }|

is bounded by \frac{1}{6} ||π(φ_{2})||_{ℓ^{1}}, so that if

∑\{π(φ_{1})(τxσ_{0}) : τx reduced \} < \frac{1}{2} ||π(φ_{1})||_{ℓ^{1}} = \frac{1}{2} ||π(φ_{2})||_{ℓ^{1}}

then

∑\{π(φ_{2})(τxσ_{0}) : τx reduced \} < \frac{1}{2} ||π(φ_{2})||_{ℓ^{1}} + \frac{1}{6} ||π(φ_{2})||_{ℓ^{1}} = \frac{4}{6} ||π(φ_{2})||_{ℓ^{1}} = \frac{2}{3} ||π(φ_{2})||_{ℓ^{1}}.

□

Definition An equivalence relation E on a standard Borel space (X, B) has modular type if there is a countable group G acting by Borel automorphisms on X with E = E_{G} and there is an increasing sequence of finite Boolean algebras

B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ ... ⊂ B_{n} ⊂ B_{n+1} ⊂ ...B,

such that G’s action permutes each B_{n} and B is generated as a σ-algebra by

⋃_{n∈ℕ}B_{n}.

In particular then we have X, ∅ ∈ B_{0} and if A is an atom in some B_{n} and g ∈ G, then g · A is still an atom in B_{n}.

Example Consider the space 2^{ℕ} =_{df} {f|f : ℕ → {0, 1}}, equipped with the metric d(f_{1}, f_{2}) = 2^{−n(f_{1},f_{2})}, where n(f_{1}, f_{2}) is the least n with f_{1}(n) ≠ f_{2}(n). Suppose, as in the examples constructed by Ted Slaman and John Steel, [19], we have an action of 𝔽_{2} by isometries on this space relative to this metric: That is to say, for all f_{1}, f_{2} ∈ 2^{ℕ} and σ ∈ 𝔽_{2} we have d(f_{1}, f_{2}) = d(σ · f_{1}, σ · f_{2}).

Then for each s ∈ 2^{<ℕ} a finite binary sequence, we let

V_{s} = {f ∈ 2^{ℕ} : f ⊃ s}.

Taking B_{n} = {V_{s} : length(s) ≤ n} we have witnesses to modularity.

We will in future refer to this space with this action as X_{ss}.

Alexander Kechris has suggested a different way of thinking about these classes of examples. If we order the atoms of the various B_{n}’s under inclusion, thereby obtaining a tree structure, then we can identify the space with a subset of the infinite branches, and view the group as acting by isomorphisms. In this way modular actions can be identified with actions by automorphisms of a rooted tree.

In the next theorem we equip 2^{𝔽_{2}} =_{df} {0, 1}^{𝔽_{2}} with the product measure, μ, and the shift action on the right: (τ · f)(σ) = f(στ). We let E_{𝔽_{2}} denote this equivalence relation.

Theorem 3.6 Let E be of modular type on (X, B) and M ⊂ 2^{𝔽_{2}} of full measure. Then there is no countable to one measurable

θ : M → X

such that for all f_{1}, f_{2} ∈ 2^{𝔽_{2}}

f_{1}E_{𝔽_{2}}f_{2} ⇒ θ(f_{1})Eθ(f_{2}).

Proof Note first of all that we may assume that on some positive measure set M_{0} ⊂ M we have that θ|_{M_{0}} is one to one. There are various ways to see this: One approach is to appeal to the uniformization theorem for subsets of the plane with countable sections, and argue that θ[M] is Borel and admits a Borel right inverse ρ : θ[M] → M; the image of ρ has conull saturation, and hence positive measure; since θ ∘ ρ(y) = y all y ∈ Dom(ρ) we can simply take the image of ρ as our positive set.

For the time being I wish to make a drastic simplifying assumption: θ is injective everywhere. This will ease some of the notation, and we can return to the further problems faced in the general case after having seen the main ideas. In actual fact it will turn out that the general argument is only slightly more involved and requires only minor regearing.

Let G be a countable group acting by Borel automorphisms with E = E_{G}, B_{0} ⊂ B_{1} ⊂ ... ⊂ B_{n} ⊂ B_{n+1} ⊂ ...B with ⋃ B_{n} generating B and each B_{n} a finite G-invariant algebra. We let A(B_{n}) denote the atoms of the algebra B_{n}.

For B ∈ B we let

B̂ = θ^{−1}[B].

We can then define B̂ to be {B̂ : B ∈ B}, B̂_{n} = {B̂ : B ∈ B_{n}}, and A(B̂_{n}) to be the atoms of B̂_{n}. Note that off of a measure zero set we still have B̂ generating the Borel algebra of 2^{𝔽_{2}}. (See 15.2 of [11]. This is the place where we are using θ injective.)

For x ∈ {a, a^{−1}, b, b^{−1}}, f ∈ 2^{𝔽_{2}}, let g_{f,x} ∈ G be such that

g_{f,x} · θ(f) = θ(x · f).

Note that this assignment f ↦ g_{f,x} can be chosen to be measurable, for any such x. Claim(I): ∀x ∈ {a, a^{−1}, b, b^{−1}}∀ϵ > 0∃M ⊂ 2^{𝔽_{2}}∃n ∈ ℕ such that μ(M) > 1−ϵ and ∀B̂ ∈ A(B̂_{n})∀f_{1}, f_{2} ∈ B̂ ∩ M

g_{f_{1},x} = g_{f_{2},x};

in other words, the function f ↦ g_{f,x} depends only on which B̂ ∈ A(B̂_{n}) the point f resides. Proof of Claim: One approach is to fix F ⊂ G finite such that off of a set of measure less than ϵ/2 we have each g_{f,x} ∈ F; we then go on and let (M_{g})_{g∈F} be such that for all f ∈ M_{g}

g_{f,x} = g

and μ(⋃_{g∈F} M_{g}) > 1 − ϵ/2. Since ⋃ B̂_{n} generates B̂ as a σ-algebra, we have that ⋃_{n∈ℕ} B̂_{n} is dense in the measure algebra. Thus for each g ∈ 𝔽 we may choose n_{g} ∈ ℕ and B̂_{g} ∈ B̂_{n_{g}} with

μ(B̂_{g}ΔM_{g}) < \frac{ϵ}{2|F |} .

Thus taking

n = max{n_{g} : g ∈ F }

and

M = ( ⋃_{g∈F} M_{g}) \ ( ⋃_{g∈F} B̂_{g}ΔM_{g})

we are done. (Claim□)

Applying this last claim repeatedly we may produce sets N_{2} ⊂ N_{1} ⊂ 2^{𝔽_{2}}, n ∈ ℕ, and

g : A(B̂_{n}) × {a, a^{−1}, b, b^{−1}} → G,

(B̂, x) ↦ g_{B̂,x},

such that (i) for all τ ∈ 𝔽_{2} with^{2} d(τ, e) ≤ 3 and all f ∈ N_{2} we have τ · f ∈ N_{1};

(ii) μ(N_{2}) > 1 − (10^{−9});
(iii) if B̂ ∈ A(B̂_{n}), f ∈ B̂ ∩ N_{1}, x ∈ {a, a^{−1}, b, b^{−1}}, then
g_{B̂,x} = g_{f,x}.

We may further assume that μ(B̂) < 10^{−8} all B̂ ∈ A(B̂_{n}). Claim(II): If x ∈ {a, a^{−1}, b, b^{−1}}, B_{1}, B_{2} ∈ A(B_{n}), and x · [B̂_{1} ∩ N_{1}] ∩ B̂_{2} ∩ N_{1} ≠ 0, then

(i) x · [B̂_{1} ∩ N_{1}] ⊂ B̂_{2};

(ii) x^{−1} · [B̂_{2} ∩ N_{1}] ⊂ B̂_{1};
(iii) x · [B̂_{1}] ⊃ B̂_{2} ∩ N_{1}.

Proof of Claim: (i): Choose f_{0} ∈ B̂_{1} ∩ N_{1} ∩ x^{−1} · [B̂_{2}]. Then g_{f_{0},x} = g_{B̂_{1},x}, and since

g_{B̂_{1},x} · θ(f_{0}) = θ(x · f_{0}) ∈ B_{2}

and since G permutes A(B_{n}) we have g_{B̂_{1},x} · B_{1} = B_{2}. And thus for all f ∈ B̂_{1} ∩ N_{1} we have θ(x · f) = g_{f,x} · θ(f) = g_{B̂_{1},x} · θ(f) ∈ B_{2},

∴ x · f ∈ B̂_{2}.

(ii): Choose f_{0} ∈ B̂_{1} ∩ N_{1} ∩ x^{−1} · [B̂_{2} ∩ N_{1}]. Letting f_{1} = x · f_{0} we have f_{1} ∈ B̂_{2} ∩ N_{1},

x^{−1} · f_{1} ∈ B̂_{1},

and we finish by applying (i) but with x^{−1} taking the place of x and B̂_{1} and B̂_{2} exchanging roles we are done. (iii): If f_{0} ∈ B̂_{2} ∩ N_{1}, then x^{−1} · f_{0} ∈ B̂_{1} by (ii). (Claim□)

2We write d(τ_{0}, τ_{1}) to indicate the distance of τ_{0} from τ_{1} in the Cayley graph; thus d(τ, e) ≤ 3 indicates that τ can be written as a word in {a, a^{−1}, b, b^{−1}, e} of length at most 3.

Definition Let us say that B̂ ∈ A(B̂_{n}) is good if

\frac{μ(B̂ \ N_{2})}{μ(B̂)} < 10^{−4};

if it is not good then we say it is bad.

It follows from μ(N_{2}) > 1 − (10^{−9}) that

μ(⋃{B̂ ∈ A(B̂_{n}) : B̂ is bad} ∪ (2^{𝔽_{2}} \ N_{2}) < 10^{−4}.

The remainder of the proof has the following overarching form. The last claim more or less states that 𝔽_{2} comes close to permuting the good elements of A(B̂_{n}). But then if we look at the centers associated to the characteristic functions of the good elements of A(B̂_{n}) we obtain a set which is overly 𝔽_{2}-invariant, and a contradiction ensues.

Let

N_{3} = {f ∈ 2^{𝔽_{2}} : ∀τ ∈ 𝔽_{2}(d(τ, e) ≤ 3 ⇒ τ · f ∈ N_{2} \ ⋃{B̂ ∈ A(B̂_{n}) : B̂ bad})}.

Note that μ(N_{3}) > 1 − (4 × 3 × 3 × 10^{−4}) > 1 − 10^{−2}. Claim(III): If f_{0} ∈ N_{3} ∩ B̂_{1}, τ ∈ 𝔽_{2}, d(τ, e) ≤ 3, τ · f_{0} ∈ B̂_{2}, then

\frac{μ(B̂_{1})}{μ(B̂_{2})} ∈ ( \frac{10^{4} − 1}{10^{4}} , \frac{10^{4}}{10^{4} − 1} )

and

μ(τ · B̂_{1}ΔB̂_{2}) < 4 · 10^{−4}min{μ(B̂_{1}), μ(B̂_{2})}.

Proof of Claim: Let us suppose τ = x_{0}x_{1}x_{2}, each x_{i} ∈ {a, a^{−1}, b, b^{−1}}; let us fix Ĉ_{1}, Ĉ_{2} ∈ A(B̂_{n}) containing x_{2} · f and x_{1}x_{2} · f. Then by (iii) of the last claim, x_{2} · B̂_{1} ⊃ Ĉ_{1} ∩ N_{1}, x_{1} · Ĉ_{1} ⊃ Ĉ_{2} ∩ N_{1}, and x_{0}Ĉ_{2} ⊃ B̂_{2} ∩ N_{1}. But for any f′ ∈ B̂_{2} ∩ N_{2} we have

τ^{−1} · f′ = x_{2}^{−1}x_{1}^{−1}x_{0}^{−1} · f′, x_{1}^{−1}x_{0}^{−1} · f′, x_{0}^{−1} · f′ ∈ N_{1};

tracking backwards we obtain x_{0}^{−1} · f′ ∈ Ĉ_{2} ∩ N_{1}, x_{1}^{−1}x_{0}^{−1} · f′ ∈ Ĉ_{1} ∩ N_{1}, and finally τ^{−1} · f′ ∈ B̂_{1}. And thus, bearing in mind that τ · f ∈ B̂_{2} implying B̂_{2} good, we in general obtain

τ · [B̂_{1}] ⊃ B̂_{2} ∩ N_{2}
∴ (μ(B̂_{2} \ (τ · B̂_{1})) < 10^{−4}μ(B̂_{2})
∴ μ(B̂_{2}) − μ(B̂_{1}) = μ(B̂_{2}) − μ(τ · B̂_{1}) < 10^{−4}μ(B̂_{2}).

Similar reasoning implies τ^{−1} · [B̂_{2}] ⊃ B̂_{1} ∩ N_{2},

∴ B̂_{2} ⊃ τ · [B̂_{1} ∩ N_{2}]
∴ μ((τ · B̂_{1}) \ B̂_{2}) < 10^{−4}μ(B̂_{1}).

And then as in the previous paragraph we have

∴ μ(B̂_{1}) − μ(B̂_{2}) < 10^{−4}μ(B̂_{1}).
(Claim□)

Definition For B̂ ⊂ 2^{𝔽_{2}} with μ(B̂) ∈ (0, 1) we define γB̂ ∈ L^{2}(2^{𝔽_{2}}) by

γB̂(f) = \frac{√(1 − μ(B̂))}{√(μ(B̂))}

for f ∈ B̂ and

γB̂(f) = \frac{−√(μ(B̂))}{√(1 − μ(B̂))}

for f ∉ B̂. Note then that

||γB̂||_{L^{2}}^{2} = ∫ (γB̂)^{2}

= ∫B̂ \frac{1 − μ(B̂)}{μ(B̂)} + ∫_{2^{𝔽_{2}} \B̂} \frac{μ(B̂)}{1 − μ(B̂)} = 1

and that

⟨γB̂, 1⟩ = ∫B̂((1 − μ(B̂))/μ(B̂))^{\frac{1}{2}} − ∫_{2^{𝔽_{2}} \B̂}(μ(B̂)/(1 − μ(B̂)))^{\frac{1}{2}}
= √(μ(B̂))√(1 − μ(B̂)) − √(1 − μ(B̂))√(μ(B̂)) = 0.

We therefore have a unit vector which is orthogonal to the constant functions.

We say that σ_{0} is a center (or weak center) for f ∈ 2^{𝔽_{2}} if for B̂ ∈ A(B̂_{n}) containing f we have that σ_{0} is a center (respectively, weak center) for π(γB̂), the weight associated to the element of A(B̂_{n}) in which f falls.

Claim(IV): If f ∈ N_{3}, τ ∈ 𝔽_{2}, d(τ, e) ≤ 3, f ∈ B̂_{1}, τ · f ∈ B̂_{2}, B̂_{1}, B̂_{2} ∈ A(B̂_{n}), then

||τ · γB̂_{1} − γB̂_{2}||_{L^{2}} = ||γ_{τ·B̂_{1}} − γB̂_{2}||_{L^{2}} < 3 · 10^{−2}.

Proof of Claim: We may partition 2^{𝔽_{2}} into the sets τ · B̂_{1}ΔB̂_{2}, τ · B̂_{1} ∩ B̂_{2}, 2^{𝔽_{2}} \ (τ · B̂_{1} ∪ B̂_{2}) and let ψ_{1}, ψ_{2}, ψ_{3} be the functions with support equal to those sets respectively and such that

γ_{τ·B̂_{1}} − γB̂_{2} = ψ_{1} + ψ_{2} + ψ_{3}.

It suffices to find suitable bounds on ||ψ_{1}||_{L^{2}}, ||ψ_{2}||_{L^{2}}, ||ψ_{3}||_{L^{3}}. For ease of expression we assume μ(B̂_{1}) ≤ μ(B̂_{2}); this is harmless; the argument under the reverse inequality is symmetrical. Recall that we have

μ(B̂_{2}) < \frac{10^{4}}{10^{4} − 1} μ(B̂_{1})

by the last claim.

We calculate upper bounds on ||ψ_{1}||_{L^{2}}, ||ψ_{2}||_{L^{2}}, ||ψ_{3}||_{L^{2}} in turn.

||ψ_{1}||_{L^{2}} = (∫_{τ·B̂_{1}ΔB̂_{2}} ψ_{1}^{2})^{1/2} ≤ [(4 · 10^{−4}μ(B̂_{1}))( \frac{1}{√(μ(B̂_{1}))} )^{2})^{\frac{1}{2}} = √(4 · 10^{−4}) = 2 · 10^{−2}.

||ψ_{2}||_{L^{2}} = (∫_{τ·B̂_{1}∩B̂_{2}} ψ_{2}^{2})^{\frac{1}{2}} ≤ √(μ(B̂_{2}))[ \frac{√(1 − μ(B̂_{1}))}{√(μ(B̂_{1}))} − \frac{√(1 − μ(B̂_{2}))}{√(μ(B̂_{2}))} ]

≤ √(μ(B̂_{2}))[ \frac{√(1 − μ(B̂_{1}))}{√(μ(B̂_{1}))} − \frac{√(1 − μ(B̂_{2}))}{√(μ(B̂_{2}))} ][ \frac{√(1 − μ(B̂_{1}))}{√(μ(B̂_{1}))} + \frac{√(1 − μ(B̂_{2}))}{√(μ(B̂_{2}))} ](√(B̂_{2}))

= μ(B̂_{2})[ \frac{(1 − μ(B̂_{1}))}{(μ(B̂_{1}))} − \frac{(1 − μ(B̂_{2}))}{(μ(B̂_{2}))}

= μ(B̂_{2})( \frac{μ(B̂_{2}) − μ(B̂_{1})μ(B̂_{2}) − μ(B̂_{1}) + μ(B̂_{1})μ(B̂_{2})}{μ(B̂_{1})μ(B̂_{2})}

= \frac{μ(B̂_{2}) − μ(B̂_{1})}{μ(B̂_{1})} < \frac{10^{−4}μ(B̂_{1})}{μ(B̂_{1})} = 10^{−4}.

||ψ_{3}||_{L^{2}} = ((∫_{2^{𝔽_{2}} \(τ·B̂_{1}∪B̂_{2})}[ \frac{√(μ(B̂_{1}))}{√(1 − μ(B̂_{1}))} − \frac{√(μ(B̂_{2}))}{√(1 − μ(B̂_{2}))} ]^{2})^{\frac{1}{2}} ≤ [( \frac{√(μ(B̂_{1}))}{√(1 − μ(B̂_{1}))} )^{2} + ( \frac{√(μ(B̂_{2}))}{√(1 − μ(B̂_{2}))} )^{2}]^{\frac{1}{2}}

< (2μ(B̂_{1}) + 2μ(B̂_{2}))^{\frac{1}{2}} < (2 · 10^{−8} + 2 · 10^{−8})^{\frac{1}{2}} = 2 · 10^{−4}.

(Claim□)

Claim(V): If f ∈ N_{3}, τ ∈ 𝔽_{2}, d(τ, e) ≤ 3, σ_{0} a center for f, then σ_{0}τ^{−1} is a weak center for τ · f. Proof of Claim: We fix B̂_{1}, B̂_{2} ∈ A(B̂_{n}) with f ∈ B̂_{1} and τ · f ∈ B̂_{2}. By definition we then have σ_{0} a center for π(γB̂_{1}). By (iv) 3.5 we have that σ_{0} · τ^{−1} is a center for π(τ · γB̂_{1}) = π(γ_{τ·B̂_{1}}). By claim(IV) we have ||γ_{τ·B̂_{1}} − γB̂_{2}||_{L^{2}} < \frac{1}{12} , and hence by (v) 3.5 σ_{0}τ^{−1} is a weak center for π(γB̂_{2}) and hence by definition a weak center for τ · f.

Definition For x ∈ {a, a^{−1}, b, b^{−1}} we let A_{x} be the set of f ∈ N_{3} such that either the identity e is a center of f or some reduced word of the form σx is a center of f. For τ a reduced word we let B_{τ} be the set of f ∈ 2^{𝔽_{2}} such that some reduced στ is a weak center for f.

Each f ∈ N_{3} is a member of at least one A_{x} by lemma 3.1, and hence we may fix A_{x} with μ(A_{x}) > \frac{1}{5} . From the last claim we have that if f ∈ A_{x} ∩ N_{3}, and τ does not begin with x, and d(τ, e) ≤ 3, then

τ · f ∈ B_{xτ^{−1}} ⊂ B_{τ^{−1}}.

There are 27 possibilities for a reduced word τ of length 3 not starting with x, and for each such τ we have

μ(B_{xτ^{−1}}) ≥ μ(A_{x}) > \frac{1}{5} .

Since \frac{27}{5} > 2 there is some f ∈ 2^{𝔽_{2}} in three different B_{xτ_{1}^{−1}}, B_{xτ_{2}^{−1}}, B_{xτ_{3}^{−1}}, each xτ_{i} reduced, each d(τ_{i}, e) = 3. This flatly contradicts lemma 3.3.

So much for the proof of the theorem under the simplifying assumption that θ is one to one everywhere. In the general case we must first pass to M_{0} of positive measure on which θ is injective. Since the

action of 𝔽_{2} on 2^{𝔽_{2}} is mixing we may find invertible mpts

T, S : M_{0} → M_{0}

such that at each x ∈ M_{0} there will be n(x), m(x) > 0 with

T (x) = a^{n(x)} · x,
S(x) = b^{m(x)} · x,

and thus we have ⟨T, S⟩ ≅ 𝔽_{2}.

Now we trot through the argument above, but selectively replacing ⟨a, b⟩ with ⟨T, S⟩. We still calculate the centers with respect to the Cayley graph on 𝔽_{2} = ⟨a, b⟩, but we use only words from ⟨T, S⟩ to shift.

We amend the definitions from above by setting B̂ = M_{0} ∩ θ^{−1}[B] for B ∈ B; as in claim(I) we may find an n such that for all B̂ ∈ A(B̂_{n}), all x ∈ {T, T^{−1}, S, S^{−1}}, and f_{1}, f_{2} in a subset of M_{0} with relatively large measure, g_{f_{1},x} = g_{f_{2},x}, where f ↦ g_{f,x} has been chosen measurably with the requirement that θ(x.f) = g_{f,x} · θ(x).

Our definitions of N_{2}, N_{1} are parallel to the ones before, but now asking that N_{2} ⊂ N_{1} ⊂ M_{0} and μ(N_{2}) > (1 − 10^{−8})μ(M_{0}) and having x range over {T, T^{−1}, S, S^{−1}}. In claim(II) we have x again range over {T, T^{−1}, S, S^{−1}} and in claim(III) we have τ range over words of length less than three built from {T, T^{−1}, S, S^{−1}}. The main idea here is that if τ is an irreducible word in {T, T^{−1}, S, S^{−1}} and f ∈ M_{0}, then there will be a corresponding τ̄_{f} built from {a, a^{−1}, b, b^{−1}} with τ̄_{f} · f = τ · f and various other natural properties: σ = τ if and only if σ̄_{f} = τ̄_{f}; σ and τ are incomparable (i.e. neither extends the other) if and only if σ̄_{f} is incomparable with τ̄_{f}.

With these and other natural adjustments the proof passes through as before. □

Kechris pointed out a more elegant approach to the last step of the argument, which passes from one to one somewhere to the specific case of one to one everywhere. We begin by assuming we have a homomorphism^{3} θ from E_{𝔽_{2}} as in the statement of the theorem; we are aiming for a contradiction. We then choose Borel ρ : θ[M] × ℕ ↠ M such that {ρ(y, n) : n ∈ ℕ} enumerates the preimage of any y ∈ θ[M]. We let G×ℤ act on X ×2^{ℕ} with the product of the original action of G on X and the odometer action of ℤ on 2^{ℕ} (the generator acts by adding one with carry, but in fact any modular action of ℤ or any other countably infinite group will do); this action will still be modular since the odometer action of ℤ is modular. We let (z_{n})_{n∈ℕ} enumerate some orbit in 2^{ℕ} under the action of ℤ. We now define a new homomorphism

θ̂: M → X × 2^{ℕ}

given by

θ̂(x) = (θ(x), z_{n}),

where n is least such that ρ(θ(x), n) = x. This new homomorphism is one to one everywhere, and falls into the proof above.

There is one obvious generalization which the argument accommodates.

Definition Let (Z, C, ν) be a standard Borel probability space and let Z^{𝔽_{2}} be the product space consisting of all

f : 𝔽_{2} → Z

with the product measure and the action defined by

σ · f(τ) = f(τσ).

This space with this action and measure is said to be the Bernoulli shift of 𝔽_{2} on Z^{𝔽_{2}}.

Theorem 3.7 Let E be of modular type on (X, B), (Z, C, ν) a standard Borel probability space, and M ⊂ Z^{𝔽_{2}} of full measure. Then there is no countable to one measurable

θ : M → X

such that f_{1}E_{𝔽_{2}}f_{2} ⇒ θ(f_{1})Eθ(f_{2}).

Proof We let (C_{ℓ})_{ℓ∈ℕ} be a generating algebra of ν-independent sets in Z, each having measure one half. For S ⊂ ℕ × 𝔽_{2} finite we define

ψ_{S} : Z^{𝔽_{2}} → {−1, 1}
f ↦ (−1)^{|{(ℓ,σ)∈S:f(σ)∈C_{ℓ}}|}.

With this and other painless modifications the proof goes through as before. □

4 Borel reducibility

Definition For E and F equivalence relations on standard Borel (X, B) and (Y, C), we say that E is Borel reducible to F, written E ≤_{B} F, if there is a Borel function θ : X → Y with

x_{1}Ex_{2} ⇔ θ(x_{1})F θ(x_{2}).

We write E <_{B} F if E ≤_{B} F holds but F ≤_{B} E fails.

3A Borel function θ is said to be a homomorphism from an equivalence relation F to an equivalence relation F′ if x_{1}F x_{2} always implies θ(x_{1})F′θ(x_{2}).

We say that E is treeable if it is countable and there is an acyclic graph on X, which is Borel in the sense of having its collection of vertices Borel as a subset of X × X in the product Borel structure, whose connected components form the E equivalence classes. E is hyperfinite if it can be written as an increasing union of Borel equivalence relations which have all equivalence classes finite.

Any free Borel action of 𝔽_{2} is treeable; one simply copies the Cayley graph across the various components of E_{𝔽_{2}}. An equivalence relation with countable classes is hyperfinite if and only if it is Borel reducible to E_{0}. A group is amenable if and only if it has a free action by measure preserving transformations on a standard Borel probability space whose resulting orbit equivalence relation is hyperfinite.

For these and other related facts the reader should refer to [13].

Theorem 4.1 (Jackson, Kechris, Louveau; see [13].) There is a universal treeable equivalence relation; that is to say, there is a treeable equivalence relation E_{T ∞} such that for all F with countable classes we have

F ≤_{B} E_{T ∞}

if and only if F is treeable.

Not all treeable equivalence relations are universal in their sense. E_{0} is treeable, but not ≤_{B}-universal. But in answer to a question raised in [13]:

Theorem 4.2 There is an equivalence relation E on standard Borel (X, B) with

E_{0} <_{B} E <_{B} E_{T ∞}.

Proof Let E be a modular equivalence relation, arising from a free ergodic action of 𝔽_{2} as in [19], as discussed following the original definition of modular in the previous section. Since it arises from a free Borel action of 𝔽_{2}, it is treeable; since there is an invariant measure and 𝔽_{2} is non-amenable, it is not Borel reducible to E_{0}. To see that it is not ≤_{B}-universal among treeable equivalence relations we apply 3.6. □

John Clemens pointed out a further application of 3.6. He begins by considering the equivalence relation E_{𝔽_{2}}^{X_{ss}} × E_{0}, where, as before, 𝔽_{2} acts in a measure preserving fashion on X_{ss} with modular type. This resulting product equivalence relation is still modular, and hence in particular E_{T ∞} is not Borel reducible to E_{𝔽_{2}}^{X_{ss}} × E_{0}; on the other hand by [1] the product equivalence relation is not treeable and thus we also have E_{𝔽_{2}}^{X_{ss}} × E_{0} not Borel reducible to E_{T ∞}.

In this way we have an example of a countable Borel equivalence relation which is not ≤_{B}-comparable with E_{T ∞}. As Clemens puts it, E_{T ∞} is not a node among the countable Borel equivalence relations.

There is one manner in which the definition of modular could be relaxed. Instead of asking that each B_{n} be finite we ask only that it be countable and generated as an algebra by its atoms.

Definition An equivalence relation E on a standard Borel space (X, B) is loosely modular if there is a countable group G acting by Borel automorphisms on X with E = E_{G} and there is an increasing sequence of countable algebras

B_{0} ⊂ B_{1} ⊂ ... ⊂ B_{n} ⊂ ...B

such that (a) each B ∈ B_{n} is a finite Booelan combination of the atoms in B_{n};

(b) each B_{n} is permuted by the action of G.

However in the context of equivalence relations and actions on finite measure spaces this generalization, although painless, buys relatively little new. We leave the following proposition as an exercise for the reader.

Proposition 4.3 Let E on (X, B) be loosely modular, F an ergodic equivalence relation on a standard Borel probability space (Y, C, μ), and

θ : Y → X

a countable to one measurable map with

y_{1}F y_{2} ⇒ θ(y_{1})Eθ(y_{2}).

Then there is a modular Ê on (X̂, B̂) and a countable to one measurable θ̂ : Y → X̂ with y_{1}F y_{2} ⇒ θ̂(x_{1})Êθ̂(x_{2}).

Thus if F refuses measurable reduction to a modular equivalence relation then it likewise refuses measurable reduction to any loosely modular equivalence relation.

An example of a loosely modular equivalence relation is given by any countable group of permutations of ℕ acting by right composition on the space of injections from ℕ to some countable set. It follows then in particular that there are countable Borel equivalence relations which are not reducible to say the group of recursive permutations acting on the space of injections ℕ ↪ ℕ.

Since the first draft of this paper, Kechris in [12] has written up his own treatment of this material which draws out the representation theoretic connections and in which he formalizes the notion of antimodular.

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