The complexity of the classification of Riemann surfaces and complex manifolds ^{∗}

G. Hjorth^{†} and A.S. Kechris^{†}

Abstract In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of ℂ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalence relations.

On the other hand we show that the analog of Becker, Rubel, and Henson’s question has a negative solution in (complex) dimension n ≥ 2.

1 Introduction

In this paper we consider the problem of classifying various classes of complex manifolds. The investigation is completely abstract, since we are not so much concerned with specific schemes of classification and their success or failure, but rather in considering what kinds of complete invariants could in principle be produced. Here we have in mind that there is a hierarchy of levels of difficulty of classifying various mathematical objects and this paper joins Dougherty-Jackson-Kechris [94], Harrington-Kechris-Louveau [90], Hjorth [9?a], Hjorth-Kechris [95], Kechris [92], Kechris [98], and others, as one more piece in a general project to compare the classification problems across a variety of mathematical disciplines and obtain a language that can contrast their various forms.

At perhaps the simplest level are schemes of classification which provide a single point in some highly concrete space as a complete invariant. In ergodic theory the real number corresponding to the entropy of a Bernoulli shift is a complete invariant for this class of measure preserving transformations. One similarly finds in the theory of Riemann surfaces that compact complex surfaces considered up to conformal equivalence may be cataloged as points in highly concrete spaces.

∗AMS Subject classification: 04A15, 03A15, 32C10; Key words and phrases: Borel equivalence relations, classification problems, complex manifolds, Riemann surfaces, conformal equivalence

†Research partially supported by NSF Grants DMS 96-22977 (G.H.) and DMS 96-19880 (A.S.K.)

The work below came about after seeing the following theorem:

Theorem 1.1. (Becker-Rubel-Henson [80]) There is no “reasonably concrete space” X and “reasonably intrinsic” or “reasonably definable” assignment

f : D → X

from D, the collection of complex domains, such that for any two R_{1}, R_{2} ∈ D

R_{1} ≅ R_{2} ⇔ f(R_{1}) = f(R_{2}).

In other words, there is no reasonable way to assign points in X as complete invariants for complex domains considered up to conformal equivalence.

Of course the phrases “reasonably concrete space” and “reasonably intrinsic assignment” are deliberately vague. For the purposes of this introduction we simply ask that the reader accept that this can and will be made precise, that the various ways in which one might do so are not subject to serious controversy, and that all the competing explications of these phrases give rise to similar outcomes. Indeed while one may have doubts about how a definition of such concepts should be crafted, it should certainly be clear that for instance the space of all subsets of ℂ is not sufficiently concrete and that a function obtained by invoking the axiom of choice to well order the complex domains and then thus armed producing an injection from D into the ordinals or into ℂ or into P_{ℵ_{0}}(ℂ) (the set of all countable subsets of ℂ) can not be considered “reasonably intrinsic” or “reasonably definable”.

Perhaps in passing we can mention that for us “reasonably concrete space” means something like a Polish space or a standard Borel space, and that for us a ”reasonably definable function” means something like a function that is Borel measurable from some standard Borel space of parametrizations. More generously one may consider, as in the Ulm invariants from abelian group theory (see Kaplansky [69]), spaces such as the set of all countable subsets of the first uncountable ordinal, or equivalently, countable transfinite sequences from ℂ, and functions that are universally Baire measurable, or projective, or even ordinal definable from reals. As we discuss in section 5.C, Theorem 1.1 survives in some form even in these contexts.

Our first concern is the extent to which classification may be obtained if we consider invariants more general than a single point in some space. In direct response to a question by Becker, Rubel, and Henson, we show that countable unordered subsets of ℂ can provide complete invariants for R, the class of all Riemann surfaces:

Theorem 1.2. There is a ”reasonably definable” function

f : R → P_{ℵ_{0}}(ℂ),

from the Riemann surfaces to countable subsets of ℂ, such that for any two R_{1}, R_{2}

R_{1} ≅ R_{2} ⇔ f(R_{1}) = f(R_{2}).

In other words, there is a “reasonably definable” way to assign countable subsets of a concrete space as complete invariants for Riemann surfaces considered up to conformal equivalence.

One might draw an analogy between this result and the Halmos-von Neumann [42] invariants for discrete spectrum ergodic measure preserving transformations. There, as here, there is no reasonable method to assign points in say ℂ as complete invariants, but the countable subset of ℂ corresponding to the eigenvalues completely classifies an ergodic discrete spectrum measure preserving transformation up to isomorphism. The proof of 1.2 in 4.A below is based on an abstract method and does not seem to provide “geometrically meaningful” invariants f(R). One can wonder if it is possible to sharpen this theorem by providing such invariants.

The method of proof in 4.A below actually gives a much better upper bound on the complexity of conformal equivalence for Riemann surfaces and in 4.B we show that this is precise.

Theorem 1.3. The classification problem for Riemann surfaces considered up to conformal equivalence is “equal in difficulty” to that of the universal countable Borel equivalence relation, E_{∞}.

Here “equal in difficulty” indicates that each equivalence relation can be embedded in the other using a function that is Borel measurable in some suitable space of parameters. E_{∞} is known to have a number of instantiations. For instance, it can be realized as the orbit equivalence relation produced by the shift action of F_{2} (the free group on 2 generators) on 2^{F_{2}} = {f|f : F_{2} → {0, 1}}, or as isomorphism on finitely branching trees (Jackson-Kechris-Louveau [9?]), or (very recently, Thomas-Velickovic [98]) isomorphism on finitely generated groups.

Put another way, this means that the “moduli space” of all Riemann surfaces is “Borel equivalent” to the very complicated quotient space 2^{F_{2}}/E_{∞}. Actually, as it follows from the proof of 1.3 given in section 4 below, this holds as well for Riemann surfaces homeomorphic to the infinitely punctured plane, i.e., ℂ \ S, where S ⊆ ℂ is infinite discrete. Thus this gives a precise measure of the set theoretic complexity of the moduli space of these Riemann surfaces. It is much more complex than the moduli spaces of finitely punctured compact Riemann surfaces, which are fairly “concrete” and admit a rich geometrical structure.

Finally, the higher dimensional case is discussed in section 6. Here we provide a new lower bound on the complexity of biholomorphism, and indicate a sense – which when translated into the theory of Borel equivalence relations can be made totally precise – in which the passage from complex dimension 1 to complex dimension 2 brings an increase in classification difficulty. The following is a simple corollary of 6.1 below.

Theorem 1.4. Let M^{2} be the class of two dimensional complex manifolds. Then there is no “reasonably definable” assignment

f : M^{2} → P_{ℵ_{0}}(ℂ)

such that for any M_{1}, M_{2} ∈ M^{2},

M_{1} ≅ M_{2} ⇔ f(M_{1}) = f(M_{2}).

2 Borel equivalence relations

Definition 2.1. A topological space is said to Polish if it is separable and admits a complete metric. We then define the Borel sets to be those appearing in the smallest σ-algebra containing the open sets. A function between two Polish spaces is said to be Borel if the inverse image of any open set under f is Borel.

If X is a Polish space and E is an equivalence relation, then E is said to be Borel if it is Borel as a subset of X × X (in the product topological structure). For x ∈ X, we let [x]_{E} = {y ∈ X : xEy}. An equivalence relation is countable if every [x]_{E} is countable.

Examples of Polish spaces

(i) ℝ, ℂ, in their usual topologies; 2^{ℕ} =_{df} {f|f : ℕ → {0, 1}} equipped with

the metric

d(x, y) = 2^{− least n such that x(n)≠y(n)}

is a compact metric space, and so certainly the underlying topological space is Polish.

(ii) Polish spaces are closed under countable products, and thus we obtain that ℝ^{n}, ℂ^{n}, ℝ^{ℕ}, ℂ^{ℕ}, ℕ^{ℕ} are all Polish.

(iii) Any G_{δ} subset of a Polish space is Polish (see Kechris [95, 3C]).

(iv) The space of all countable models of a given countable language is a Polish space; this is an especially important example from the point of view of logic. Let L be a countable (relational) language and let Mod(L) be the set of all L-structures with underlying set ℕ and equipped with the topology induced by taking as subbasic open sets those of the form

{M ∈ Mod(L) : M ⊧ R(n_{1}, ⋯, n_{k})},
{M ∈ Mod(L) : M ⊧ ¬R(n_{1}, ⋯, n_{k})},

for n_{1}, ⋯, n_{k} in ℕ and R ∈ L a relation of arity k. This space is Polish since it is homeomorphic to

∏_{R∈L} 2^{ℕ^{a(ℝ)}},

where a(R) is the arity of the relation symbol R (see Becker-Kechris [96]). (A similar definition works even for languages with function symbols, since we can show that the space Mod(L) is homeomorphic to a G_{δ} subset of Mod(L′) for a suitably chosen relational L′.)

Many mathematical objects can be thought of as points in some appropriately chosen Polish space considered up to some notion of isomorphism or equivalence. For instance, we may think of countable groups as elements of some appropriate Mod(L) considered up to isomorphism. A fundamental notion used in comparing equivalence relations is that of Borel reducibility.

Definition 2.2. For E and F equivalence relations on Polish spaces X and Y, we write E ≤_{B} F, and say that E is Borel reducible to F, to indicate that there is a Borel function f : X → Y such that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ f(x_{1})F f(x_{2}).

We use E ∼_{B} F for E ≤_{B} F ≤_{B}; E <_{B} F indicates that E ≤_{B} F but it is not the case that F ≤_{B} E.

Examples of equivalence relations

(i) id(ℂ), the equality relation on ℂ; more generally for any Polish space X we can consider the identity relation id(X) on X. Since any two uncountable Polish spaces are Borel isomorphic, one obtains for any two uncountable Polish spaces X and Y that id(X) ∼_{B}id(Y).

(ii) We let E_{0} be the equivalence relation of eventual agreement on 2^{ℕ}. This means that x_{1}E_{0}x_{2} if and only if there is some N ∈ ℕ such that for all m > ℕ, x_{1}(m) = x_{2}(m). Here it is well known that id(ℂ) <_{B} E_{0} (see Hjorth [9?a, Chapter 3], or Hjorth-Kechris [95], or Kechris [98]).

(iii) Given a countable group G, we can let it act on the Polish space 2^{G}(=_{df} {f|f : G → {0, 1}} in the product topology) by shift: for g ∈ G and f ∈ 2^{G} we define g · f by

(g · f)(h) = f(g^{−1}h)

for h ∈ G. We denote the orbit equivalence by E(G, 2).

(iv) In specific cases the complexity under ≤_{B} of E(G, 2) is exactly understood. For instance it is known that

E_{0} ∼_{B} E(ℤ, 2)

(Weiss, Slaman-Steel; see Dougherty-Jackson-Kechris [94]) and that for any Borel equivalence relation E one has either

E ≤_{B} id(ℂ)

or

E_{0} ≤_{B} E,

but not both (this was proved in Harrington-Kechris-Louveau [90]).

(v) Let F_{2} be the free group on 2 generators. Here it is known that for any countable Borel equivalence relation E

E ≤_{B} E(F_{2}, 2).

In this sense, we say that E(F_{2}, 2) is a universal countable Borel equivalence relation, and we will in the future denote it by E_{∞}.

(vi) More generally, we consider orbit equivalence relations induced by the continuous actions of Polish groups (that is, topological groups whose underlying space is Polish). If G is a Polish group acting continuously on a Polish space X we will use E_{G}^{X} for the orbit equivalence relation given by

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

(vii) Finally, for L a countable language, we let ≅_{Mod(L)} be the equivalence relation of isomorphism on elements of Mod(L). Thus M ≅_{Mod(L)} N iff there is some permutation π of the natural numbers such that for all relations R in the language

M ⊧ R(n_{1}, ⋯, n_{k}) ⇔ N ⊧ R(π(n_{1}), ⋯, π(n_{k})).

Definition 2.3. We say that an equivalence relation E is smooth if E ≤_{B} id(X) for some Polish space X; note that here we may assume without loss of generality that X = ℂ. We say that E admits classification by countable structures if there is some countable language L with

E ≤_{B}≅_{Mod(L)} .

Let us further say that E admits classification by countable subsets of ℂ if there is a countable sequence of Borel functions (f_{n})_{n∈ℕ} into ℂ such that for all x_{1}, x_{2}

x_{1}Ex_{2} ⇔ {f_{n}(x_{1}) : n ∈ ℕ} = {f_{n}(x_{2}) : n ∈ ℕ}.

Note that in the definition of classification by countable subsets we ask that {f_{n}(x_{1}) : n ∈ ℕ} and {f_{n}(x_{2}) : n ∈ ℕ} are equal as unordered sets; if we were to alternatively require something along the lines of the equality of the sequences (f_{n}(x_{1}))_{n∈ℕ} and (f_{n}(x_{2}))_{n∈ℕ} then this would be nothing other than the notion of smoothness, since ℂ^{ℕ} is itself a Polish space.

The next couple of lemmas clarify these definitions.

Lemma 2.4. If E is a countable Borel equivalence relation, then E admits classification by countable subsets of ℂ.

Proof. Let X by the Polish space on which E is defined. Following the uniformization theorem for Borel sets in the plane with countable sections (see Kechris [95, 18.C]) we may find a countable sequence of Borel functions

h_{n} : X → X

such that for all x ∈ X

[x]_{E} = {h_{n}(x) : n ∈ ℕ}.

Then if h : X ↪ ℂ is a Borel injection we may let f_{n} = h ∘ h_{n} witness the definition at 2.3 ⊣

This implication does not reverse (see for instance, Hjorth [9?a, §2.3]).

Lemma 2.5. If E admits classification by countable subsets of ℂ, then E admits classification by countable structures.

Proof. Let (P_{n})_{n∈ℕ} be a sequence of unary predicates, {U_{n} : n ∈ ℕ} a countable basis for the topology of ℂ, and let

⟨·, ·⟩ : ℕ^{2} → ℕ

be a bijection. Let (f_{n})_{n∈ℕ} be the sequence of Borel functions witnessing the classifiability by countable subsets of ℂ. Let L be the countable language determined by (P_{n})_{n∈ℕ}.

Then for any point x in the space on which E is defined we let M_{x} be the L-model with underlying set ℕ defined by

M_{x} ⊧ P_{n}(⟨k, l⟩)

if and only if f_{k}(x) ∈ U_{n}. (Here each f_{k}(x) is quite deliberately repeated infinitely often; allowance needs to be made for the fact that the sets {f_{n}(x_{1}) : n ∈ ℕ} and {f_{n}(x_{2}) : n ∈ ℕ} may be equal, but the sequences (f_{n}(x_{1}))_{n∈ℕ} and (f_{n}(x_{2}))_{n∈ℕ} may list a given complex number with different frequency.) ⊣

The implication here also fails to reverse (see, for example, Hjorth-Kechris-Louveau [98]).

Although we confine ourselves to Borel functions, a similar reducibility theory arises for any general class of functions including the Borel, closed under composition, and satisfying reasonable regularity properties (such as Lebesgue measurability and the pullbacks of open sets having the Baire property). This is an important point. Key theorems are not held hostage to circumstantial and subjective choices used in our explication of reasonably definable.

For instance we could equally work with the C-measurable functions, i.e., those that pull back open sets to sets lying in the smallest σ-algebra containing the open sets and closed under Souslin’s A-operation (compare Hjorth-Kechris [95]). Alternatively we might use the absolutely Δ_{2}^{1} functions.

For functions of higher complexity, it is necessary to work in systems stronger than ZFC to avoid the whole pursuit dissolving into a quick sand of indistinguishable independence results. For instance, we can develop a similar theory for functions in L(ℝ) (the smallest class of sets containing all the reals and closed, in some sense, under transfinite iterations of rudimentarily definable operations) assuming sufficient determinacy or the existence of large cardinals. The authors of Becker-Henson-Rubel [80] choose to use the collection of all “ordinal definable functions” and in this context one again obtains a similar theory by working in the very specific model of set theory known as the Solovay model.

3 Parametrizing complex manifolds

In order to precisely formulate the problem of classifying species of complex manifolds we need a method of parametrizing complex manifolds. Here the method must be completely general, and in some sense given in advance of any specific classification result. We will use points in a standard Borel space – indeed without loss of generality we can take the space ℝ – to describe or parametrize complex manifolds. The obvious properties will become Borel in the space of parametrizations and the equivalence relation of biholomorphism will be Σ_{1}^{1} (analytic) – that is to say, defined by the Borel image of a Borel set. It should not be felt that there is any great mystery or presumption in the method of parameterization. It is an empirical fact that most classes of concrete mathematical objects can be represented by points in a standard Borel space and all such methods of representation tend to be “Borel equivalent”.

We recount some basic definitions, all detailed in Kodaira [86].

Definition 3.1. A complex manifold of dimension n is a connected Hausdorff second countable topological space M along with a chart {U_{α}, φ_{α}}_{α∈Λ}, where:

(i) ⋃_{α∈Λ} U_{α} = M;

(ii) each φ_{α} : U_{α} → V_{α} is a homeomorphism between U_{α} ⊆ M, open in M, and V_{α} ⊆ ℂ^{n}, open in ℂ^{n};

(iii) the overlap maps

φ_{β} ∘ φ_{α}^{−1} : φ_{α}(U_{α} ∩ U_{β}) → φ_{β}(U_{α} ∩ U_{β}), when U_{α} ∩ U_{β} ≠ ∅,

are holomorphic.

A complex manifold of dimension 1 is called a Riemann surface. For M and N complex manifolds, with charts {U_{α}, φ_{α}}, {W_{β}, ψ_{β}}, we say that τ : M → N is holomorphic if each ψ_{β}τφ_{α}^{−1} is holomorphic. A biholomorphic map is a holomorphic map with holomorphic inverse. We shall write

M ≅ N

if they are biholomorphically equivalent, that is to say, there is some biholomorphic

π : M → N.

For n = 1, i.e., for Riemann surfaces, one usually says that M, N are conformally equivalent if M ≅ N.

A manifold is a “separable” object, and thus we might hope to represent or parametrize a manifold by describing the arrangement of a countable dense subset. There is some judgment regarding the kind of objects to take as satisfactory parameterizations; plainly the class of all manifolds would be absurdly complicated and the set of all subsets of say ℂ×ℂ^{2} ×ℂ^{3} … only slightly less so. Our idea of a reasonable space is something like ℝ, or the set of all subsets of ℕ, or C([0, 1]), or infinite strings from a finite alphabet, or any of the classical Banach spaces. In each case we are less interested in the topological structure of these spaces than their Borel structure.

Definition 3.2. A set S equipped with a σ-algebra B of subsets is said to be a standard Borel space if there is a completely metrizable separable topology τ on S which gives rise to B as the σ-algebra generated by its open sets.

As a remark on terminology, we say that a completely metrizable separable topological space is Polish, and that the Borel sets for a Polish topology are those appearing in the smallest σ-algebra containing its open sets. Thus we may say that (S, B) is a standard Borel space if there is a Polish topology τ on S that has B as its Borel sets.

As well as the usual examples of Polish spaces, such as ℝ^{n}, ℂ^{n}, 2^{N}, stripped down to their Borel structure, we should also mention that given any Polish space X the collection of all closed subsets, F(X), is a standard Borel space, when equipped with the σ-algebra generated by sets of the form {F ∈ F(X) : F ∩ U ≠ ∅} for U ⊆ X open; this more subtle example is known as the Effros standard Borel space and is discussed in §12.C of Kechris [95]. It follows that the space of all open subsets of X is a standard Borel space in the σ-algebra generated by sets of the form {O ⊆ X open : O ∩ U = ∅}.

The collection of standard Borel spaces is closed under taking countable disjoint unions and countable products. In consequence of §13.A in Kechris [95], any Borel subset of a standard Borel space is again standard Borel in the induced subspace Borel structure.

We can then proceed to define for each n ≥ 1, a standard Borel space

M^{n}

and a map

p ∈ M^{n} ↦ M_{p}

assigning to each p ∈ M^{n} an n-dimensional complex manifold M_{p} such that for every n-dimensional complex manifold M there is at least one p ∈ M^{n} with M ≅ M_{p}. We call M^{n} the parameter space of n-dimensional complex manifolds. The construction of M^{n} and the verification that it has a number of reasonable properties that we will need in various parts of this paper is technically cumbersome, although mathematically rather shallow. We will thus postpone the precise definition and verification of these facts until §7. In our proofs between now and then we will simply state as lemmas the various facts about this coding that we need and return to their proofs in §7.

To start off, the following fact gives an upper bound for the complexity of biholomorphic equivalence.

Proposition 3.3. For each n ≥ 1, the relation

p ≅_{n} q ⇔ p, q ∈ M^{n} and M_{p} ≅ M_{q}

is Σ_{1}^{1}.

For n = 1 we will denote M^{1} by R and ≅_{1} by ≅_{R} (for Riemann surfaces) and we will see in §4 that actually ≅_{R} is Borel, i.e., the relation of conformal equivalence of Riemann surfaces is Borel (in the parameters). However we doubt if this is true in higher dimensions.

Conjecture 3.4. For n ≥ 2, the biholomorphic equivalence relation ≅_{n} of ndimensional complex manifolds (in the parameters) is not Borel.

We will also discuss in this paper a particular class of Riemann surfaces, namely domains (open connected sets) in ℂ. Again we will need to parametrize these by elements of a standard Borel space. We will thus define in §7 a standard Borel space D and a map d ∈ D ↦ D_{d} assigning to each d ∈ D a domain D_{d} ⊆ ℂ such that for each domain D ⊆ ℂ there is at least one d ∈ D with D = D_{d}. We call D the parameter space of domains in ℂ. Of course for every domain D there is also a p ∈ R such that M_{p} ≅ D. The next fact asserts that such a p can be computed in a Borel way from any d ∈ D such that D = D_{d}.

Proposition 3.5. There is a Borel function f : D → R such that D_{d} ≅ M_{f(d)}.

In particular if we denote by ≅_{D} the relation of conformal equivalence (in the parameters), i.e.,

d ≅_{D} e ⇔ d, e ∈ D & D_{d} ≅ D_{e}

then we have that

(≅_{D}) ≤_{B} (≅_{R}).

4 Riemann surfaces

Our main goal in this section is to compute the precise complexity of conformal equivalence of Riemann surfaces and planar domains in the hierarchy of Borel equivalence relations.

Theorem 4.1. The conformal equivalence relations ≅_{R}, ≅_{D} (in the parameters) of Riemann surfaces and planar domains are Borel and

(≅_{R}) ∼_{B} (≅_{D}) ∼_{B} E_{∞}.

We will split the proof of this theorem in two parts:

The upper bound, i.e., showing that ≅_{R} is Borel and (≅_{R}) ≤_{B} E_{∞}.

The lower bound, i.e., showing that E_{∞} ≤_{B} (≅_{D}).

Since, as we already pointed out in §3, (≅_{D}) ≤_{B} (≅_{R}) this will complete the proof of 4.1.

4.A The upper bound

We will need here some basic facts from the Uniformization Theory of Riemann Surfaces (see, e.g. Forster [81]).

Given a Riemann surface M, we will devote by M̂ its universal covering Riemann surface and by π : M̂ → M the covering map. Thus (M̂, M, π) has the following properties:

(i) M̂ is simply connected and π is a holomorphic surjection of M̂ onto M;

(ii) π evenly covers M, i.e., for each x ∈ M there is open U containing x, so that π^{−1}[U] is the disjoint union of open sets {V_{i}}, where each π|V_{i} : V_{i} → U is a biholomorphism.

(iii) M̂ is uniquely, up to biholomorphism, determined by (i), (ii) above, i.e., if f : M → N is a biholomorphism and (N̂, N, ρ) satisfy (i), (ii) above, then there is a biholomorphism f̂: M̂ → N̂ such that ρ ∘ f̂ = f ∘ π. Moreover, for each x̂_{0} ∈ M̂, ŷ_{0} ∈ N̂ with f(π(x̂_{0})) = ρ(ŷ_{0}), there is a unique f̂ as above with f̂(x̂_{0}) = ŷ_{0}.

The following fundamental result is variously called the Uniformization Theorem or the Riemann Mapping Theorem for Riemann Surfaces.

Theorem 4.2. Every simply connected Riemann surface is conformally equivalent to exactly one of the following:

(i) ℂ_{∞} = the Riemann Sphere (= ℂ ∪ {∞});

(ii) ℂ = the complex plane;

(iii) ℍ = {x + iy : y > 0} = the upper half plane.

The group of automorphisms (i.e., biholomorphic correspondences) of a Riemann surface M will be denoted by Aut(M). It turns out that the automorphism groups of the simply connected Riemann surfaces can be explicitly described as follows (see, e.g., Beardon [84], Bedford et al. [91]).

(i) Aut(ℂ_{∞}) = P SL_{2}(ℂ)

Here P SL_{2}(ℂ) is the quotient of the group SL_{2}(ℂ) of all complex matrices (^{a} ^{b}cd) with ad − bc = 1 by its center {+I, −I}, where I is the identity matrix. This acts on ℂ_{∞} as a Möbius transformation, i.e., for (^{a} ^{b}cd) ∈ P SL_{2}(ℂ), z ∈ ℂ̂

(\begin{matrix}a & b \\ c & d \end{matrix}) (z) = \frac{az + b}{cz + d}

(ii) Aut(ℂ) = the “az + b” group.

This is the group of all pairs (a, b) ∈ ℂ^{∗} × ℂ, when ℂ^{∗} = ℂ − {0}, with multiplication (a, b) · (a′, b′) = (aa′, ab′ + b). It acts on ℂ by (a, b)(z) = az + b.

(iii) Aut(ℍ) = P SL_{2}(ℝ)

Here P SL_{2}(ℝ) is the quotient of the group SL_{2}(ℝ) of all real matrices (^{a} ^{b}cd) with ad − bc = 1 by its center {+I, −I}. This again acts on ℍ by Möbius transformations.

In particular, these automorphism groups are Lie groups, therefore Polish locally compact, and their actions on the underlying spaces are continuous.

Now consider a Riemann surface M and a universal covering π : M̂ → M, where we now take M to be one of ℂ_{∞}, ℂ, ℍ. Let

F_{π} = {(x, y) ∈ M̂: π(x) = π(y)}.

Then F_{π} is a closed subset of M̂^{2}, i.e., F_{π} ∈ F(M̂^{2}). Now let f : M → N be a biholomophic map and ρ : N̂ → N a universal covering. Then N̂ = M̂ and there is σ ∈ Aut(M̂) such that π = ρ ∘ σ. Thus

(x, y) ∈ F_{ρ} ⇔ ρ(x) = ρ(y)
⇔ π (σ^{−1}(x)) = π (σ^{−1}(y))
⇔ (σ^{−1}(x), σ^{−1}(y)) ∈ F_{π}
⇔ (x, y) ∈ σ · F_{π},

where Aut(M̂) acts on F(M̂^{2}) by

σ · F = {(x, y) : (σ^{−1}(x), σ^{−1}(y)) ∈ F }

Thus F_{ρ}, F_{π} belong to the same orbit of this action.

Conversely, suppose π : M̂ → M, ρ : M̂ → N are two universal coverings and for some σ ∈ Aut(M̂), F_{ρ} = σ · F_{π}. Then we can define f : M → N by

f(x) = ρ (σ(y)) , where π(y) = x.

Notice that this is well-defined, since if π(y) = π(y′) = x, i.e., (y, y′) ∈ F_{π}, then (σ(y), σ(y′)) ∈ F_{ρ}, so ρ (σ(y)) = ρ (σ(y′)). It is now clear that f is a biholomorphism between M, N.

Thus we have seen that fixing M̂ ∈ {ℂ_{∞}, ℂ, ℍ} there is an injection from the set of conformal equivalence classes of Riemann surfaces with universal covering surface M̂ into the orbits of the action of Aut(M̂) on F(M̂^{2}). We have thus essentially reduced our problem to the study of this action. But then we can use the following general result:

Theorem 4.3. (Kechris [92]) Let G be a Polish locally compact group, X a standard Borel space, (g, x) ↦ g · x a Borel action of G on X. Then there is a Borel set S ⊆ X which meets every orbit in a countable nonempty set.

This implies the following, concerning the orbit equivalence relation E_{G}^{X}:

xE_{G}^{X}y ⇔ ∃g ∈ G(g.x = y).

First, it is well-known that E_{G}^{X} is Borel (see Kechris [95, 35.49]). Thus, E_{G}^{X}|S is countable Borel and satisfies (E_{G}^{X}|S) ≤_{B} E_{∞}. Also there is a Borel function f : X → S with f(x)E_{G}^{X}x, for all x ∈ X (see Kechris [95, 18C]). Thus E_{G}^{X} ≤_{B} E_{G}^{X}|S, so

E_{G}^{X} ≤_{B} E_{∞}

In particular, this applies to the action of Aut(M̂) on F(M̂^{2}), where M̂ ∈ {ℂ_{∞}, ℂ, ℍ}, since this is clearly a Borel action.

One way to proceed then is to show the following:

(i) For each M̂ ∈ {ℂ_{∞}, ℂ, ℍ}, the set UM̂ = {z ∈ R : M_{z} has universal covering surface M̂} is Borel.

(ii) There is a Borel function F : UM̂ → F(M̂^{2}) so that F (z) is of the form F_{π}, for some covering π : M̂ → M.

Then letting EM̂ be the orbit equivalence relation induced by the action of Aut(M̂) on f(M̂^{2}), we have that for z, w ∈ UM̂,

z ≅_{R} w ⇔ M_{z} ≅ M_{w}
⇔ F (z)EM̂F (w),

i.e., (≅_{R}) ≤_{B} EM̂ and since EM̂ ≤_{B} E_{∞} we have that (≅_{R}) ≤_{B} E_{∞}.

The drawback of this approach is that the construction of the function F in (ii) above, although intuitively rather clear, involves some messy computations, which we don’t want to commit to print. So we will follow an alternative, somewhat indirect, approach that will minimize the technicalities.

We will use the following criterion (a proof of which can be found in Hjorth [9?a, 5.2]).

Proposition 4.4. (Kechris) For a Borel equivalence relation E on a standard Borel space X, the following are equivalent:

(i) E ≤_{B} E_{∞},

(ii) There is a Borel function f : X → Y, with Y a standard Borel space, such that for any x, y ∈ X:

(a) f([x]_{E}) is countable;

(b) ¬xEy ⇒ f([x]_{E}) ∩ f([y]_{E} = ∅.

Put M̂_{1} = ℂ_{∞}, M̂_{2} = ℂ, M̂_{3} = ℍ. Consider the action of Aut(M̂_{i}) on F(M̂_{i}^{2}) and let E_{i} be the corresponding orbit equivalence relation. Let also S_{i} be Borel sets that meet every E_{i}-class in a countable nonempty set. Then there are Borel functions f_{i,j}, i = 1, 2, 3, j = 1, 2, ⋯, f_{i,j} : F(M̂_{i}^{2}) → F(M̂_{i}^{2}) such that {f_{i,1}(F) : j = 1, 2, ⋯ } = S_{i} ∩ [F]_{E_{i}} (see Kechris [95, 18.C]). We can of course assume that f_{i,1}(F) = F for F ∈ S_{i}. Thus if we put f_{i} = f_{i,1}, f_{i} satisfies (ii) of 4.4 for E_{i}. Now define the following relation P ⊆ R × Y, where Y = S_{1} ∪ S_{2} ∪ S_{3}, and we of course consider S_{1}, S_{2}, S_{3} as pairwise disjoint:

P (z, y) ⇔ ∃i ∈ {1, 2, 3}∃π : M̂_{i} → M_{z} (π is a covering map and y = f_{i}(F_{π})).

Denote by P_{z} the section of P determined by z ∈ R. Then if z_{1} ≅_{R} z_{2} and y_{1} ∈ P_{z_{1}}, y_{2} ∈ P_{z_{2}}, say y_{1} = f_{i}(F_{π_{1}}), y_{2} = f_{i}(F_{π_{2}}) (for some i ∈ {1, 2, 3, }), then clearly F_{π_{1}}E_{i}F_{π_{2}}, so y_{1} ∈ f_{i}([F_{π_{1}}]_{E_{i}}) = f_{i}([F_{π_{2}}]_{E_{i}}) ∋ y_{2}. Thus P_{z_{1}}, P_{z_{2}} ⊆ f_{i}([F_{π_{1}}]_{E_{i}}). It follows that for each z ∈ R, ⋃_{w≅_{R}z} P_{w} = P̄_{z} is countable.

If now P̄_{z_{1}} ∩ P̄_{z_{2}} ≠ ∅ and y ∈ P̄_{z_{1}} ∩ P̄_{z_{2}}, then y = f_{i}(F_{π_{1}}) = f_{i}(F_{π_{2}}), where i ∈ {1, 2, 3} and π_{1} : M̂_{i} → M_{z′_{1}}, π_{2} : M̂_{i} → M_{z′_{2}} are covering maps with z′_{1} ≅_{R} z_{1}, z′_{2} ≅_{R} z_{2}. Then F_{π_{1}}E_{i}F_{π_{2}}, so z′_{1} ≅_{R} z′_{2}, and thus z_{1} ≅_{R} z_{2}.

So we have:

(i) z ∈ R ⇒ P̄_{z} is countable,

(ii) z_{1} ≅_{R̸} z_{2} ⇒ P̄_{Z_{1}} ∩ P̄_{Z_{2}} = ∅.

We can now complete the proof by showing the following two facts:

(a) ≅_{R} is Borel;

(b) P is Borel.

Indeed, from (b) and the fact that every section of P is countable, we can find a Borel function f : R → Y such that f uniformizes P, i.e.

P (z, f(z))

for any z ∈ R. Then clearly f satisfies condition (ii) of 4.4 and so by (a) above we can apply 4.4 to conclude that (≅_{R}) ≤_{B} E_{∞}.

Proof of (a): We need the following lemma concerning our parameterization, which will be proved in §7.

Lemma 4.5. Let T_{i} ⊆ R × F(M̂_{i}^{2}) be defined by

(z, F) ∈ T_{i} ⇔ ∃π : M̂_{i} → M_{z} (π is a covering map and F = F_{π}).

Then T_{i} is Σ_{1}^{1}.

It follows that if C_{i} ⊆ R is defined by

z ∈ C_{i} ⇔ the universal cover of M_{z} is M̂_{i},

then C_{i} = proj_{R}(T_{i}), so C_{i} ∈ Σ_{1}^{1} as well. Since {C_{1}, C_{2}, C_{3}} is a partition of R, it follows that actually each C_{i} is Borel.

We have already seen in §2 that ≅_{R} is Σ_{1}^{1}, so to show that it is Borel it is enough to show that it is Π_{1}^{1}. Recall that the equivalence relations E_{i} induced by the action of Aut(M̂_{i}) on F(M̂_{i}^{2}) are Borel. Then we have:

z_{1} ≅_{R} z_{2} ⇔ ∃i ∈ {1, 2, 3}[z_{1}, z_{2} ∈ C_{i} & ∀F_{1}∀F_{2}[(z_{1}, F_{1}) ∈ T_{i} & (z_{2}, F_{2}) ∈ T_{i} ⇒

F_{1}E_{i}F_{2}]

which shows that ≅_{R} is Π_{1}^{1}.

Proof of (b): We have that

P (z, y) ⇔ ∃i ∈ {1, 2, 3}[(z, F) ∈ T_{i} & f_{i}(F) = y],

so clearly P is Σ_{1}^{1}. To see that it is also Π_{1}^{1}, notice that

P (z, y) ⇔ ∃i ∈ {1, 2, 3}[z ∈ C_{i} & ∀F ((z, F) ∈ T_{i} ⇒ ∃j(f_{i}(f_{i,j}(F)) = y))].

4.B. The lower bound

We will use the following realization of E_{∞}:

Recall that we let F_{2} be the free group with 2 generators and let p(F_{2}) be the set of all subsets of F_{2} with the topology given by its identification with 2^{F_{2}}, the latter having the product topology. Consider the left-translation action of F_{2} on p(F_{2})

g · A = gA,

which is clearly continuous. Remember that we denote by E(F_{2}, 2) the associated orbit equivalence relation and we identify E(F_{2}, 2) with E_{∞}. It is therefore enough to show that

E(F_{2}, 2) ≤_{B} (≅_{D}).

To do this we will associate to each A ⊆ F_{2} a discrete subset S_{A} ⊆ ℍ and consider the domain D_{A} = ℍ \ S_{A}. We will show that

A E(F_{2}, 2)B ⇔ D_{A} ≅ D_{B}

(where D_{A} ≅ D_{B} means of course that D_{A}, D_{B} are conformally equivalent). Moreover the construction is very explicit, so that there is a Borel function f : p(F_{2}) → ℍ^{ℕ} such that for any A ⊆ F_{2}, f(A) enumerates S_{A}. We will prove in §7 the following easy fact about our parameterization of domains.

Proposition 4.6. There is a Borel function g : ℍ^{ℕ} → D so that for any x ∈ ℍ^{ℕ} with {x_{n} : n ∈ ℕ} discrete we have that D_{g(x)} = ℍ \ {x_{n} : n ∈ ℕ}.

Then if h = g ∘ f we have that

A E(F_{2}, 2)B ⇔ h(A) ≅_{D} h(B)

i.e., E(F_{2}, 2) ≤_{B} (≅_{D}).

We will now proceed to the construction of A ↦ S_{A}.

Consider the group P SL(2, ℤ) of all integer matrices in P SL(2, ℝ) = Aut(ℍ). This acts properly discontinuously on ℍ, thus each orbit of this action is a discrete subset of ℍ (see, e.g., Katok [92]). Moreover, there are subgroups of P SL(2, ℤ) isomorphic to F_{2} which have no fixed points under this action. An example is the group generated by the two generators

σ = (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) , τ = (\begin{matrix}1 & 0 \\ 2 & 1 \end{matrix})

(see Wagon [93, p. 61, 7.1]). We will identify from now on F_{2} with the subgroup ⟨σ, τ⟩ of P SL(2, ℤ) generated by σ, τ. Then for any fixed x ∈ ℍ, {g(x) : g ∈ F_{2}} is discrete and g(x) = x iff g = 1 (the identity of F_{2}).

So fix from now on x_{1} ∈ ℍ and put x_{g} = g(x). Then g(x_{h}) = x_{gh}.

Next we consider the hyperbolic metric ρ on ℍ given by

ρ(z, w) = ln \frac{|z − w̄| + |z − w|}{|z − w̄| − |z − w|}.

It of course also induces the usual topology on ℍ. The main fact that we will use is that each element of Aut(ℍ) = P SL(2, ℝ) acts by isometries on (ℍ, ρ) (see again Katok [92]). Since {x_{g} : g ∈ F_{2}} is discrete, there is an ϵ > 0 such that {y ∈ ℍ : ρ(x_{1}, y) < ϵ} ∩ {x_{g} : g ∈ F_{2}} = {x_{1}}. It follows that for any g ∈ F_{2}, {y ∈ ℍ : ρ(x_{g}, y) < ϵ} ∩ {x_{g} : g ∈ F_{2}} = {x_{g}}.

Next put x_{1}^{(0)} = x_{1} and choose three points x_{1}^{(1)}, x_{1}^{(2)}, x_{1}^{(3)} in ℍ, distinct from each other and from x_{1}^{(0)}, such that ρ(x_{1}^{(0)}, x_{1}^{(i)}) < \frac{ϵ}{5} and the hyperbolic distances ϵ_{ij} = ρ(x_{1}^{(i)}, x_{1}^{(j)}), 0 ≤ i ≠ j ≤ 3, are all different from each other, except for ϵ_{ij} = ϵ_{ji}. Then define x_{g}^{(i)}, for i = 0, 1, 2, 3, by

x_{g}^{(i)} = g(x_{1}^{(i)}),

(so that x_{g}^{(0)} = x_{g}). Then ρ(x_{g}^{(i)}, x_{g}^{(j)}) = ρ(x_{1}^{(i)}, x_{1}^{(j)}). It follows that {y ∈ ℍ : ρ(y, x_{g}^{(0)}) < \frac{ϵ}{2}\} ∩ ⋃_{h∈F_{2}}\{x_{h}^{(i)} : 0 ≤ i ≤ 3\} = \{x_{g}^{(i)} : 0 ≤ i ≤ 3\} and that the set ⋃_{g∈F_{2}}{x_{g}^{(i)} : 0 ≤ i ≤ 3} is also discrete. Notice also that g(x_{h}^{(i)}) = x_{gh}^{(i)}.

Put for any A ∈ F_{2}

S_{A} = {x_{g}^{(i)} : g ∈ F_{2}, 0 ≤ i ≤ 2} ∪ {x_{h}^{(3)} : h ∈ A}.

We will show that this works.

First assume that A ∈ E(F_{2}, 2)B and let g ∈ F_{2} be such that gA = B. Then clearly g(S_{A}) = S_{B} and so g(D_{A}) = D_{B}, thus D_{A} ≅ D_{B}.

Conversely, assume that D_{A} ≅ D_{B} via the biholomorphism π. If Ω is a domain, 𝔻 the unit disk, a ∈ Ω and a function f : Ω → 𝔻 is holomorphic in Ω \ {a}, then by a classical theorem of Riemann (see, e.g., Rudin [66, 10.20]) a is a removable singularity, so f can be defined at a so that it remains holomorphic on Ω. Since ℍ, 𝔻 are conformally equivalent it follows that there are holomorphic extensions π^{+} : ℍ → ℍ, π^{−} : ℍ → ℍ of π, π^{−1}, respectively. Then π^{−} ∘ π^{+} is the identity on D_{A}, and vice versa, so (π^{+})^{−1} = π^{−}. Thus there is a biholomorphism π^{+} ∈ Aut(ℍ) extending π. For convenience we will just write π instead of π^{+} from now on. Clearly π(S_{A}) = π(S_{B}). We will next find g ∈ F_{2} such that π = g. To find this g consider π(x_{1}^{(0)}). For some g, π(x_{1}^{(0)}) = x_{g}^{(i)}, i ∈ {0, 1, 2, 3}. We want to argue that i = 0, i.e., π(x_{1}^{(0}) = x_{g}^{(0)}. So assume i ≠ 0, towards a contradiction. Let us look at π(x_{1}^{(i)}). We have that ρ(π(x_{1}^{(i)}), x_{g}^{(i)}) = ρ(π(x_{1}^{(i)}), π(x_{1}^{(0)})) = ρ(x_{1}^{(i)}, x_{1}^{(0)}) < \frac{ϵ}{5}, so as ρ(x_{g}^{(i)}, x_{g}^{(0)}) < \frac{ϵ}{5}, clearly ρ(π(x_{1}^{(i)}), x_{g}^{(0)}) < \frac{ϵ}{2} so that π(x_{1}^{(1)}) must be one of x_{g}^{(j)}, j ∈ {0, 1, 2, 3}, j ≠ i. Thus, ρ(x_{g}^{(i)}, x_{g}^{(j)}) = ϵ_{ij} = ρ(x_{1}^{(0)}, x_{1}^{(i)}) = ϵ_{0i}, so, by the choice of the ϵ_{ij}, we must have j = 0, i.e., π(x_{1}^{(0)}) = x_{g}^{(i)}, π(x_{1}^{(1)}) = x_{g}^{(0)}. Then let j ∈ {1, 2}, j ≠ i, and consider π(x_{1}^{(j)}). Again it must be one of the x_{g}^{(k)}, k ≠ i, 0. But then ϵ_{k0} = ρ(x_{g}^{(k)}, x_{g}^{(0)}) = ρ(π(x_{1}^{(j)}), π(x_{1}^{(i)})) = ρ(x_{1}^{(j)}, x_{1}^{(i)}) = ϵ_{ji}, a contradiction.

So π(x_{1}^{(0)}) = x_{g}^{(0)}. Then since ρ(π(x_{1}^{(1)}), x_{g}^{(0)}) = ρ(x_{1}^{(1)}, x_{1}^{(0)}) = ϵ_{10} we must have that π(x_{1}^{(1)}) = x_{g}^{(1)} and similarly π(x_{1}^{(2)}) = x_{g}^{(2)}. Thus π(x_{1}^{(i)}) = g(x_{1}^{(i)}), i ∈ {0, 1, 2}. Since both π, g are Möbius transformations this implies that π = g. Finally we need to show that gA = B, and by symmetry it is enough to see that gA ⊆ B. If h ∈ A, then x_{H}^{(3)} ∈ S_{A}. So g(x_{h}^{(3)}) = π(x_{h}^{(3)}) = x_{gh}^{(3)} ⊆ S_{B}, so gh ∈ B.

It is obvious from our construction that there is a Borel function f : p(F_{2}) → ℍ^{ℕ} with f(A) enumerating S_{A}, so the proof is complete. ⊣

5 Some consequences

5.A We first notice that the result in §4 answers the following question in Becker-Henson-Rubel [80]:

Q10. Is it consistent with the ZFC axioms of set theory to assume that there is no complete system of invariants Φ, definable by a formula in set theory, where Φ(G) is a countable (unordered) subset of ℂ for each domain G?

Using 4.1 and the fact that any two uncountable standard Borel spaces are Borel isomorphic, it follows that there is a Borel function f : D → ℂ and a countable Borel equivalence relation E on ℂ such that

d ≅_{R} e ⇔ f(d)Ef(e).

In particular, if to any planar domain D we assign the invariant Φ(D) = f([d]≅_{R}), for any d ∈ D with D_{d} = D, we obtain complete conformal invariants for planar domains which have the form of countable subsets of ℂ. Moreover, since the work in §4 is quite effective, one has that both f and E are actually effectively Borel, or Δ_{1}^{1}, and thus the invariant Φ(D) can be defined by an explicit simple formula in set theory. So Q10 is answered negatively in a strong form.

5.B By loosening-up somewhat a definition given in Hjorth-Kechris [95] let us say that an equivalence relation E on a standard Borel space X is Ulmclassifiable if there is a map U : X → Y^{<ω_{1}} where Y is a standard Borel space and Y^{<ω_{1}} = {f : α → Y : α < ω_{1}}, with xEy ⇔ U(x) = U(y), and such that U satisfies the following technical condition expressing the “niceness” of U:

(i) If α(x) = α^{U}(x) = domain of U(x) < ω_{1}, then for each α < ω_{1},

A_{α} = A_{α}^{U} = {x ∈ X : α(x) = α}

has the universal Baire property and the map U|A_{α} : X → Y^{α} is universally Baire measurable.

(ii) The set

R_{U} = {(x, y) ∈ X^{2} : α(x) ≤ α(y)}

has the universal Baire property in X^{2}.

Here a set has the universal Baire property if its preimage by any Borel function (on any Polish space) has the Baire property and a function is universally Baire measurable if its pre-composition with any such Borel function is Baire measurable.

Recall that E_{0} be the equivalence relation on 2^{ℕ} given by xE_{0}y ⇔ ∃n∀m ≥ n(x(m) = y(m)). Then it is a folklore fact that E_{0} is not Ulm-classifiable. To see this recall first that E_{0} is generically ergodic in the sense that any E_{0}-invariant set with the property of Baire is either meager or comeager. Assume now that U : 2^{ℕ} → 2^{<ω_{1}} verifies that E_{0} is Ulm-classifiable. By the Kuratowski-Ulam Theorem (see Kechris [95]) applied to R_{U} there is some α_{0} < ω_{1}, with A_{α_{0}} non-meager. Thus, A_{α_{0}} is comeager. Since there is a Borel embedding from Y into 2^{ℕ} and so a Borel embedding of Y^{α_{0}} into (2^{ℕ})^{α_{0}} we can assume, without loss of generality, that Y = 2. For α < α_{0}, there is then a unique i_{α} ∈ {0, 1} so that {x ∈ 2^{ℕ} : U(x)(α) = i_{α}} = B_{α} is comeager, so B = ⋂_{α<α_{0}} B_{α} is comeager, which is absurd, since B is a single E_{0}-equivalence class, thus countable.

Since E_{0} ≤_{B} E(F_{2}, 2) ∼_{B} (≅_{D}) ∼_{B} (≅_{R}), it follows that ≅_{R}, ≅_{D} are not Ulm-classifiable either, i.e., one cannot find complete conformal invariants for planar domains which take the form of countable transfinite sequences of reals, complex numbers, or members of any standard Borel space, and which can be defined in a “reasonable” way.

In Becker-Henson-Rubel [80], the authors show that in the generic model V [G], obtained by adding a Cohen real G to the universe V, there is no U : {domains} → ℂ^{<ω_{1}} which is a complete system of invariants and can be defined in V [G] by a formula in set theory with parameters in V. Their proof can be recast in our language as follows: If such a U existed, since E_{0} ≤_{B} (≅_{D}) by a Δ_{1}^{1} and thus explicitly defined function, we would have (in V [G]) a Ψ : 2^{ℕ} → 2^{<ω_{1}} definable by a formula of set theory with parameters in V, such that xE_{0}y ⇔ Ψ(x) = Ψ(y). By standard homogeneity properties of the Cohen poset it follows that Ψ(H_{1}) = Ψ(H_{2}) for any two Cohen generics H_{1}, H_{2} ∈ V [G], which is absurd if we take H_{1}, H_{2} to be the odd, even parts of G.

Also since Shelah [84] showed that it is consistent relative to ZF+DC that every set in a Polish space has the property of Baire, it follows that it is consistent with ZF that there is no map (definable or not) from planar domains into Y^{<ω_{1}}, Y a standard Borel space, which gives complete invariants for conformal equivalence.

5.C We have seen in §4 that the conformal equivalence relation on domains of the form ℍ \ S, S discrete, has exactly the complexity of E_{∞}. Of course the same holds for domains of the form 𝔻 \ S, S discrete in 𝔻 (𝔻 the disc). In some sense these are the simplest Riemann surfaces for which conformal equivalence can be so complex.

First if we restrict our attention to the compact case, a concrete classification of ≅_{R} for compact Riemann surfaces can be achieved, and we refer the reader to the classical theory of moduli spaces (see, e.g., Imayoshi-Taniguchi [92]). A similar result holds for the Riemann surfaces obtained by subtracting finitely many points from a compact Riemann surface (in particular all domains of the form ℂ \ {x_{1}, ⋯ x_{n}}) and domains of the form ℍ \ {x_{1}, ⋯, x_{n}}. Next we consider domains of the form ℂ \ S, with S a discrete subset of ℂ. Using the Picard Great Theorem, it follows that if ℂ\S, ℂ\S′ are conformally equivalent, then there is π ∈ Aut(ℂ) with π(S) = S′ and of course the converse is true as well. Thus, up to ∼_{B}, conformal equivalence on domains of the form ℂ \ S, S a discrete subset of ℂ, is the same as that of the orbit equivalence relation induced by the action

(g, S) ↦ g · S = g(S)

of Aut(ℂ) on the standard Borel space of discrete subsets of ℂ (a Borel subset of F(ℂ)). We will use this fact to show that conformal equivalence on the domains of the form ℂ \ S is actually <_{B} E_{∞}.

To see this, we will use the theory of amenability of countable Borel equivalence relations, see, e.g., Kechris [91].

As in that paper, we call a countable Borel equivalence relation E amenable if there is a map C ↦ Φ_{C} associating to each E-equivalence class C = [x]_{E} a mean Φ_{C} on C (i.e., a continuous functional Φ_{C} on l^{∞}(C), with inf(f) ≤ Φ_{C}(f) ≤ sup(f)) such that C ↦ Φ_{C} is universally measurable in the following sense: if F : X^{2} → [−1, 1] is Borel, the function G : X → [−1, 1] given by G(x) = Φ_{[x]_{E}}(F_{x}) (where F_{x} : [x]_{E} → ℝ is defined by F_{x}(y) = F (x, y)) is universally measurable. (A function is universally measurable if it is μ-measurable for any probability Borel measure μ).

Since E(F_{2}, 2) is not amenable (see Kechris [91, 2.3]) and amenability is preserved downwards under ≤_{B}, it follows that if E is amenable, then E <_{B} E_{∞}.

Denote the space of discrete subsets of ℂ by F_{d}(ℂ) and the orbit equivalence relation induced by the action of Aut(ℂ) on F_{d}(ℂ) by E_{d}. Then, by 4.3, there is a Borel set T ⊆ F_{d}(ℂ) meeting every E_{d}-equivalence class in a countable nonempty set. Then E_{d} ∼_{B} E_{d}|T so in order to show that E_{d} <_{B} E_{∞}, it is enough to show that E_{d}|T = E is amenable. Since the statement “E_{d} <_{B} E_{∞}” is clearly Π_{2}^{1}, so absolute under generic extensions, we can assume that the Continuum Hypothesis, CH, holds.

By a theorem of Mokobodzki (see Dellacherie-Meyer [83, pp. 102–108]), CH implies that there is a universally measurable shift-invariant mean Φ_{ℕ} on ℕ. This means that Φ_{ℕ}|[−1, 1]^{ℕ} : [−1, 1]^{ℕ} → [−1, 1] is universally measurable and Φ_{ℕ}((x_{0}, x_{1}, ⋯)) = Φ_{ℕ}(x_{1}, x_{2}, ⋯). Next we will use that Aut(ℂ) = the “az +b” group, is solvable, thus amenable, and so it has a Fölner sequence {K_{n}}. Thus {K_{n}} is a sequence of compact subsets of Aut(ℂ) which have the following properties, where λ is the left-invariant Haar measure on Aut(ℂ):

(i) λ(K_{n}) > 0

(ii) \frac{λ(hK_{n}ΔK_{n})}{λ(K_{n})} → 0, for each h ∈ Aut(ℂ) (see Patterson [88, 4.16]). We now define a mean on L^{∞}(Aut(ℂ), λ), i.e., a continuous linear functional Λ on this Banach space with essinf(f) ≤ Λ(f) ≤ esssup(f), as follows

Λ(f) = Φ_{ℕ} (n ↦ \frac{∫_{K_{n}} f(g)dλ(q)}{λ(K_{n})}) .

Then property (ii) of the Fölner sequence and the shift-invariance of Φ_{ℕ} imply that Λ is left-invariant, i.e., for any f as above if f_{h}(g) = f(hg), then

Λ(f) = Λ(f_{h}), ∀h ∈ Aut(ℂ).

Now let π : F_{d}(ℂ) → T be Borel such that π(x)E_{d}x, ∀x ∈ F_{d}(ℂ). Then for each E (= E_{d}|T)-equivalence class C = [x]_{E_{d}} ∩ T define a mean Φ_{C} on l^{∞}(C) by the formula

Φ_{C}(p) = Λ (g ↦ p(π(g · x)))

for any p ∈ l^{∞}(C), where g · x is the action of Aut(ℂ) on F_{d}(ℂ).

The left-invariance of Λ shows that this is well-defined independently of the choice of x in the same orbit of the action of Aut(ℂ) on F_{d}(ℂ). Next we check that C ↦ Φ_{C} is universally measurable, and thus E is amenable. Fix Borel F : T^{2} → [−1, 1]. We have to verify that G : T → [−1, 1] given by G(x) = Φ_{[x]_{E}}(F_{x}) = Λ(g ↦ F_{x}(π(g · x))) = Λ(g ↦ F (x, π(g · x))) is universally measurable. From the definition of Λ we have

G(x) = Φ_{ℕ} (n ↦ \frac{∫_{K_{n}} F (x, π(g · x))dλ(g)}{λ(K_{n})}) .

Put

φ(n, x) = \frac{∫_{K_{n}} F (x, π(g · x)dλ(g)}{λ(K_{n})}

Then φ : ℕ × T → [−1, 1] is Borel, so

ψ(x) = (n ↦ B(n, x))

is a Borel function from T into [−1, 1]^{ℕ}. Since G = Φ_{ℕ} ∘ ψ, and universally measurable functions are closed under composition, G is universally measurable.

On the other hand, let E(ℤ, 2) be the equivalence relation induced by the shift on p(ℤ) and consider the subset X ⊆ p(ℤ) defined by

X = {A ⊆ ℤ : A contains two consecutive integers.}

Then for A, B ∈ X

AE(ℤ, 2)B ⇔ ℂ \ A, ℂ \ B are conformally equivalent.

Now the map

A ∈ p(ℤ \ {∅}) ↦ A′ ∈ X

defined by

A′ = A ∪ {n + 1 : n ∈ A}

clearly has the property that

AE(ℤ, 2)B ⇔ A′E(ℤ, 2)B,

thus E(ℤ, 2)|p(ℤ) \ {∅} ≤_{B} E(ℤ, 2)|X. But it is known, see Dougherty-Jackson-Kechris [94], that

E_{0} ∼_{B} E(ℤ, 2)|p(ℤ) \ {∅},

from which it follows that E_{0} is ≤_{B} the conformal equivalence of planar domains of the form ℂ \ S, S a discrete subset of ℂ. We in fact conjecture the following.

Conjecture 5.1. The conformal equivalence relation on planar domains of the form ℂ \ S, S discrete in ℂ, is ∼_{B} E_{0}.

Notice also that, as in 5.B above, the conformal equivalence relation on planar domains of the form ℂ \ S, S discrete, is not Ulm-classifiable.

5.D As in Becker-Henson-Rubel [80, 6.8], we want to discuss some implications of our result to the conjugacy action on P SL_{2}(ℝ) = Aut(ℍ) on its discrete subgroups.

Put

S_{d} = {F ⊆ P SL_{2}(ℝ) : F is a discrete subgroup}.

Then S_{d} is Borel in F(P SL_{2}(ℝ)), so it is a standard Borel space. Consider the conjugacy action of P SL_{2}(ℝ) on S_{d}, which is clearly Borel and denote by E_{c} the corresponding orbit equivalence relation. Thus,

F E_{c}G ⇔ F, G are conjugate discrete subgroups of P SL_{2}(ℝ).

It is essentially shown in 6.8 of Becker-Henson-Rubel [80] that E_{0} ≤_{B} E_{c}. We compute again the exact complexity of E_{c}.

Theorem 5.2. E_{c} ∼_{B} E_{∞}.

Proof. By 4.3, E_{c} ≤_{B} E_{∞} and, by 4.B, E_{∞} ≤_{B} (≅_{R} |C_{3}), where C_{3} is the set of all z ∈ R with M_{z} having universal covering surface ℍ, so it is enough to show that (≅_{R} |C_{3}) ≤_{B} E_{c}. (Recall here that the universal covering space of any domain of the form ℍ \ S, S discrete, must be ℍ.)

If π : ℍ → M is the universal covering of a Riemann surface M, let G_{π} be the covering group, i.e., the group of all g ∈ Aut(ℍ) = P SL_{2}(ℝ) with π ∘ g = g. The group G_{π} ⊆ P SL_{2}(ℝ) is discrete and acts freely on ℍ. Moreover with the usual definition of ℍ/G_{π}, ℍ/G_{π} is conformally equivalent to M. Finally, if G, F ⊆ P SL_{2}(ℝ) are discrete subgroups acting freely on ℍ, then ℍ/G is conformally equivalent to ℍ/F iff G, F are conjugate in P SL_{2}(ℝ) (see Beardon [84]).

We will next need the following technical lemma about our parametrization, that we will prove in §7.

Lemma 5.3. There is Borel map φ : S_{d} → R such that if G ∈ S_{d} acts freely on ℍ, then ℍ/G is conformally equivalent to M_{φ(G)}.

Let now T ⊆ X = S_{d} ∩{G : G acts freely on ℍ} be a Borel set meeting every E_{c}-equivalence class in X at a countable nonempty set. Define Q ⊆ C_{3} × T by

(z, G) ∈ Q ⇔ φ(G) ≅_{R} z.

Thus Q is Borel and each section Q_{z} is countable, nonempty, so let F : C_{3} → T be Borel with (z, F (z)) ∈ Q. Then

z ≅_{R} w ⇔ F (z)E_{c}F (w),

so ≅_{R} |C_{3} ≤_{B} E_{c}, and we are done. ⊣

6 Complex manifolds

We will see here that for n ≥ 2 the biholomorphic equivalence relation on ndimensional complex manifolds is much more complicated than that of Riemann surfaces. We have the following result:

Theorem 6.1. For n ≥ 2 the relation of biholomorphic equivalence ≅_{n} of ndimensional complex manifolds (in the parameters) does not admit classification by countable structures.

Proof. It is enough to consider the case n = 2. The proof will be an application of the theory of turbulence, a consequence of which is the following result.

Proposition 6.2. (Hjorth [9?, 3.3.3]) Consider the Polish group G = (ℝ^{ℕ}, +) and let H ⊆ G be a proper Polishable subgroup. If H is strongly dense (i.e., for every (x_{0}, x_{1}, ⋯, x_{n}) ∈ ℝ^{n+1} there is x_{n+1}, x_{n+2} ⋯ with (x_{0}, x_{1}, ⋯, x_{n}, x_{n+1}, ⋯) ∈ H), then the equivalence relation

xE_{H}y ⇔ x − y ∈ H

does not admit classification by countable structures.

It will thus be enough to find such an H with

E_{H} ≤_{B} (≅_{2}).

Consider the unit disk 𝔻 and the group H_{𝔻} of all holomorphic functions f : 𝔻 → ℂ with pointwise addition. It is easy (using, e.g., Rudin [66, 10.27]) to see that with the topology of uniform convergence on compacts this is a Polish group. Consider the closed subgroup H_{𝔻}^{∗} of all f ∈ H_{𝔻} such that f(1/n) ∈ ℝ for all n ∈ ℕ, n ≥ 2. Define p : ℍ_{𝔻}^{∗} → ℝ^{ℕ} by p(f) = (f (\frac{1}{n+2})). Then

p is a continuous homomorphism from H_{𝔻}^{∗} into ℝ^{ℕ} and so p(H_{𝔻}^{∗}) = H is a Polishable subgroup of ℝ^{ℕ} (see Becker-Kechris [96, 1.6]) and is clearly proper (as the sequence (1, 0, ⋯) ∉ H). We next verify that H is strongly dense in ℝ^{ℕ} : if a_{0}, ⋯, a_{m−1}, ∈ ℝ, then there is a polynomial f with real coefficients for which f(\frac{1}{i+2}) = a_{i} for i < m, so

(f(\frac{1}{2}), ⋯, f(\frac{1}{m + 1}), f(\frac{1}{m + 2}), ⋯) = p(f) ∈ H.

So it is enough to show that

E_{H} ≤_{B} (≅_{2}).

For x = (x_{n}) ∈ ℝ^{ℕ} let M(x) be the following 2-dimensional complex manifold:

M(x) = (𝔻 × ℂ)\ ⋃_{n∈ℕ}\{\frac{1}{n + 2}\} × \{x_{n} + (n + 2), x_{n} + (n + 3), x_{n} + 2(n + 2)\}

with the obvious chart. The following easy fact about our parametrization will be checked in §7.

Lemma 6.3. There is a Borel function F : ℝ^{ℕ} → M^{2} such that

M(x) = M_{F (x)}

It is then enough to show that

x − y ∈ H ⇔ M(x) ≅ M(y).

If x − y ∈ H, so that y − x ∈ H let f ∈ H_{𝔻}^{∗} be such that p(f) = y − x, i.e., f is holomorphic on 𝔻 and f (\frac{1}{n+2}) = y_{n} − x_{n}. Then the map σ : M(x) → M(y) given by

σ(z, w) = (z, w + f(z))

is a biholomorphism between M(x), M(y), i.e., M(x) ≅ M(y).

Conversely assume that M(x) ≅ M(y) and σ : M(x) → M(y) is a biholomorphism. Fix z ∈ 𝔻. The map w ∈ ℂ ↦ π_{1}(σ(z, w)) ∈ 𝔻, where π_{1} : 𝔻 × ℂ → 𝔻 is the first projection function, is holomorphic and bounded, so by Liouville’s Theorem (see Rudin [66, 10.23]) it is constant, say C(z). (Strictly speaking, when z = \frac{1}{n+2} this map is only defined in the complement of a finite set, but it clearly has a holomorphic extension to all of ℂ.)

It is obvious that for each n, C (\frac{1}{n+2}) = \frac{1}{m+2} for some m. Thus the map w ∈ ℂ\{x_{n} + (n + 2), x_{n} + (n + 3), x_{n} + 2(n + 2)} ↦ π_{2}(σ(w)) ∈ ℂ \ {x_{m} + (m + 2), x_{m} + (m + 3), x_{m} + 2(m + 2)}, where π_{2} : 𝔻 × ℂ → ℂ in the second projection function, is a biholomorphism, so clearly ℂ\{n + 2, n + 3, 2(n + 2)}, ℂ\{m + 2, m + 3, 2(m + 2)} are conformally equivalent. Thus, as in 5.C, a transformation of the form z ↦ az + b of {n + 2, n + 3, 2(n + 2)}, gives us {m + 2, m + 3, 2(m + 2)} by considering distances we see that |a| = 1 and therefore a = 1 and m = n, i.e. C (\frac{1}{n+2}) = \frac{1}{n+2}.

Since C is holomorphic (being given by C(z) = π_{1}(σ(z, i))), it follows that C(z) = z for all z ∈ 𝔻.

Thus σ has the form

σ(z, w) = (z, ρ_{z}(w))

where ρ_{z} ∈ Aut(ℂ̂), i.e., ρ_{z}(w) = a(z)w + b(z). As b(z) = ρ_{z}(0) = π_{2}(σ(z, 0)) and a(z) = −b(z) + \frac{π_{2}(σ(z,i))}{i}, it follows that a, b are holomorphic. Since the

map a (\frac{1}{n+2}) w + b (\frac{1}{n+2}) maps \{x_{n} + (n + 2), x_{n} + (n + 3), x_{n} + 2(n + 2)\}

onto \{y_{n} + (n + 2), y_{n} + (n + 3), y_{n} + 2(n + 2)\} it follows that a (\frac{1}{n+2}) = 1 and

x_{n} + (n + 2) = b (\frac{1}{n+2}) + y_{n} + (n + 2), or x_{n} − y_{n} = b (\frac{1}{n+2}), so b ∈ H_{𝔻}^{∗} and x − y = p(b) ∈ H, and the proof is complete. ⊣

It is clear that the complex manifold used above are not compact. So the following is open:

Problem 6.4. What is the complexity of the biholomophic equivalence relation on 2-dimensional compact complex manifolds?

We do not even know if it is Borel or even concretely classifiable.

7 Technicalities of the parametrization

We start with some definitions.

Definition 7.1. Let Y_{0}^{n} be the set of all triples (O_{0}, O_{1}, φ) such that O_{0} and O_{1} are open sets in ℂ^{n} and φ : O_{0} → O_{1} is holomorphic. We provide Y_{0}^{n} with the Borel structure generated by sets of the form

{(O_{0}, O_{1}, φ) : U ⊆ O_{0}}
{(O_{0}, O_{1}, φ) : U ⊆ O_{1}}
{(O_{0}, O_{1}, φ) : φ(V) ⊆ U}

for U, V ⊆ ℂ^{n} open.

Fixing in advance for each n some enumeration (q_{i}^{n})_{i∈ℕ} (or just (q_{i})_{i∈ℕ} when context makes n clear) of a dense subset of ℂ^{n}, let Y_{1}^{n} the collection of 4-tuples

(O_{0}, O_{1}, (c_{i}), (d_{i})),

such that

(i) O_{0}, O_{1} are open sets in ℂ^{n}; each c_{i} is in ℂ^{n}; each d_{i} is in M_{n}(ℂ) (the space of all n × n matrices over ℂ);

(ii) for q_{i} not in O_{0} the values of c_{i} and d_{i} are both (0, 0, ⋯) (or any other fixed default value in ℂ^{n}); (iii) there exists φ : O_{0} → O_{1} analytic such that for each q_{i} ∈ O_{0}

φ(q_{i}) = c_{i},
φ′(q_{i}) = d_{i}.

Letting O(ℂ^{n}) be the standard Borel space of open subsets of ℂ^{n}, we have then that Y_{1}^{n} is a subspace of the standard Borel space O(ℂ^{n})×O(ℂ^{n})×(ℂ^{n})^{ℕ}× M_{n}(ℂ)^{ℕ}. Here O(ℂ^{n}) is understood to have the Borel structure generated by sets of the form {O ∈ O(ℂ^{n}) : U ⊆ O} for U open in ℂ^{n}. To see that this is a standard Borel space define

π : F(ℂ^{n}) → O(ℂ^{n})

by

F ↦ ℂ^{n} \ F;

this is obviously a Borel bijection with Borel inverse between F(ℂ^{n}) equipped with the Effros standard Borel structure and O(ℂ^{n}).

Lemma 7.2. Y_{1}^{n} is a standard Borel space.

Proof. By Kechris [95, 13.A] it suffices to show that Y_{1}^{n} is a Borel subset of O(ℂ^{n}) × O(ℂ^{n}) × (ℂ^{n})^{ℕ} × M_{n}(ℂ)^{ℕ}.

Let (C_{p})_{p∈ℕ} be the family of compact sets in ℂ^{n} consisting of all closures of basic open precompact sets. Note then that (O_{0}, O_{1}, (c_{i}), (d_{i})) ∈ Y_{1}^{n} if and only if for all C_{p} with C_{p} ⊆ O_{0} we have

(a) for all ϵ ∈ ℚ^{+} there exists δ ∈ ℚ^{+} such that

∀q_{i}, q_{j} ∈ C_{p}(|q_{i} − q_{j}| < δ ⇒ |d_{i} − d_{j}|, |c_{i} − c_{j}| < ϵ);

(b) for all ϵ ∈ ℚ^{+} there exists i_{0}, i_{1}, ⋯, i_{l}, a finite sequence in ℕ, and δ_{0}, δ_{1}, ⋯, δ_{l} in ℚ^{+} such that

C_{p} ⊆ ⋃_{j≤l} B_{δ_{j}}(q_{i_{j}})

and for all j ≤ l and q_{m}, q_{m′} ∈ B_{δ_{j}}(q_{i_{j}}) (the open ball of radius δ_{j} around q_{i_{j}}) we have that if

(ζ_{1}, ⋯, ζ_{n}) = (q_{m} − q_{m′}) ∈ ℂ^{n},
(ζ′_{1}, ⋯, ζ′_{n}) = (c_{m} − c_{m′}) ∈ ℂ^{n},

[ξ_{i,i′}]_{i,i′≤n} = d_{i_{j}}, then

|(ζ_{1}, ⋯, ζ_{n})[ξ_{i,i′}] − (ζ′_{1}, ⋯, ζ′_{n})|/|q_{m} − q_{m′}|

=_{df} |(∑_{i≤n} ζ_{1}ξ_{i,1}, ∑_{i≤n} ζ_{i}ξ_{i,2}, ⋯, ∑_{i≤n} ζ_{i}ξ_{i,n}) − (ζ′_{1}, ⋯, ζ′_{n})|/|q_{m} − q_{m′}| < ϵ.

Since all continuous functions on compact sets are uniformly continuous, (a) asserts that the assignments

q_{i} ↦ c_{i},
q_{i} ↦ d_{i}

extend to continuous functions on C_{p} while (b) asserts that q_{i} ↦ d_{i} indicates the derivative of q_{i} ↦ c_{i}. We leave to the reader the routine verifications that (a) and (b) describe Borel sets. ⊣

Corollary 7.3. For each n, Y_{0}^{n} is a standard Borel space.

Proof. The natural map from Y_{1}^{n} to Y_{0}^{n} that associates to each (O_{0}, O_{1}, (c_{i}), (d_{i})) the unique holomorphic function φ with

φ(q_{i}) = c_{i}
φ′(q_{i}) = d_{i}

for q_{i} ∈ O_{0} is clearly one to one and an isomorphism with respect to the relevant Borel structures. ⊣

Definition 7.4. Let X′_{0} be the space of all (y_{i,i′})_{i,i′∈ℕ} ∈ ℝ^{ℕ×ℕ} such that there exists a complete metric space with dense subset {a_{i} : i ∈ ℕ}, where for each i, i′

d(a_{i}, a_{i′}) = y_{i,i′}.

Note that X′_{0} is a Borel subspace of ℝ^{ℕ×ℕ}, since the only restrictions on the (y_{i,i′})_{i,i∈ℕ} is that they are ≥ 0 and accord with the triangle inequality, and that y_{i,i′} = 0 exactly when i = i′. For each such (y_{i,i′}) we let X(y_{i,i′}) be a complete metric space with dense subset {a_{i} : i ∈ ℕ} as indicated; the exact choice of X(y_{i,i′}) will be irrelevant in what follows.

We then let X_{0} be the collection of all (y_{i,i′}) ∈ X′_{0} such that for each i ∈ ℕ the open ball

B_{1}(a_{i})

of radius 1 around a_{i} in X(y_{i,i′}) is precompact.

Since precompactness can be phrased in terms of being ϵ-bounded for every rational ϵ > 0, this is a Borel subset of X′_{0}, and thus X_{0} is a standard Borel space. It is also seen that X(y_{i,i′}) will be locally compact for each (y_{i,i′}) in X_{0}. Conversely we have:

Lemma 7.5. If X is a locally compact separable metric space, then there is some (y_{i,i′}) in X_{0} such that X(y_{i,i′}) is homeomorphic to X.

Proof. Starting with X we may fix a complete d and an increasing union (O_{m})_{m∈ℕ} of precompact open sets so that for each m the closure of O_{m}, Ō_{m}, is included in O_{m+1} and such that

X = ⋃_{m∈ℕ} O_{m}.

Then we may choose continuous f_{m} : X → [m, m+2] such that f_{m} is constantly m on Ō_{m} and constantly m + 2 on X \ O_{m+1}. Then we may define d′ on X by

d′(a, a′) = d(a, a′) + ∑_{m∈ℕ} |f_{m}(a) − f_{m}(a′)|.

Letting {a_{i} : i ∈ ℕ} be any dense subset of X, we obtain (y_{i,i′}) in X′_{0} with

(y_{i,i′}) = d′(a_{i}, a_{i′}).

⊣

The next technical definition is rather cumbersome. In any case, we ultimately parametrize the complex manifolds in a Borel manner, and this is the only real concern. By a basic open set in ℂ^{n} we mean a ball with rational center and radius.

Definition 7.6. Let M^{n} be the space of sequences ((y_{i,i′}), (A_{i}), (ξ_{i,i′}))_{i,i′∈ℕ} (or just ((y_{i,i′}), (A_{i}), (ξ_{i,i′})) for short) such that

(i) (y_{i,i′}) ∈ X_{0};

(ii) A_{i} ∈ 2^{ℕ} (which we equip with the product topology and identify with the set of all subsets of ℕ);

(iii) ξ_{i,i′} ∈ ℂ^{n};

(iv) A_{j} corresponds to a regular open set in X(y_{i,i′}), in the sense that i ∈ A_{j} if and only if ∃δ ∈ ℚ^{+}∀i′ ∈ ℕ∀q ∈ ℚ^{+}(y_{i,i′} < δ ⇒ ∃i′′ ∈ A_{j}(y_{i′,i′′} < q));

(v) {a_{i} : i ∈ A_{j}} is guaranteed to be precompact by having diameter less than or equal to 1, in the sense that for all i, i′ ∈ A_{j},

y_{i,i′} < 1;

(vi) if we define U_{j} =_{df} (\overline{\{a_{i} : i ∈ A_{j}\}})^{∘} for each j to be the interior of the closure of \{a_{i} : i ∈ A_{j}\} and V_{j} =_{df} (\overline{\{ξ_{j,i} : i ∈ A_{j}\}})^{∘} for each j to be the interior of the closure of {ξ_{j,i} : i ∈ A_{j}}, then the assignment

a_{i} ↦ ξ_{j,i}

extends to a homeomorphism

φ_{j} : U_{j} → V_{j}

and the sets V_{j} are basic open and connected in ℂ^{n};

(vii) the overlap maps

φ_{j} ∘ φ_{i}^{−1} : φ_{i}(U_{i} ∩ U_{j}) → φ_{j}(U_{i} ∩ V_{j})

are biholomorphic; (viii) the topological space X(Y_{i,i′}) is connected;

(ix) for each i the closed set \overline{B_{1}(a_{i})} is covered by finitely many open sets of the form U_{j} as described in (vi).

For p ∈ M^{n} we will use M_{p} to denote the complex manifold of dimension n, X(y_{i,i′}), with chart {U_{i}, φ_{i}} as indicated above.

Lemma 7.7. M^{n} is a standard Borel space.

Proof. First let M_{0}^{n} be the collection of all ((y_{i,i′}), (A_{i}), (ξ_{i,i′}), (V_{i,j}), (φ_{i,j})) such that:

(i) ((y_{i,i′}), (A_{i}), (ξ_{i,i′})) satisfy (i)-(vi) and (ix) of 7.6,

(ii) for each i, j we have (V_{i,j}, V_{j,i}, φ_{i,j}) in Y_{0}^{n},

(iii) each V_{i,j} is the interior of the closure of {ξ_{i,l} : l ∈ A_{j} ∩ A_{i}},

(iv) for each l ∈ A_{j} ∩ A_{i}; we have

φ_{i,j}(ξ_{i,l}) = ξ_{j,l},

(v) for all i, i′ there is a finite sequence j_{1} = i, j_{2}, ⋯, j_{k} = i′ such that each A_{j_{n}} ∩ A_{j_{n+1}} ≠ ∅.

Note that (v) expresses that M_{p} is connected. Thus ((y_{i,i′}), (A_{i}), (ξ_{i,i′})) is in M^{n} if and only if there is some (V_{i,j}), (φ_{i,j}) with ((y_{i,i′}), (A_{i}), (ξ_{i,i′}), (V_{i,j}), (φ_{i,j})) in M_{0}^{n} and the functions

φ_{j} ∘ φ_{i}^{−1} : φ_{i}(U_{i} ∩ U_{j}) → φ_{j}(U_{i} ∩ U_{j})

are equal to φ_{i,j}.

M_{0}^{n} is a subset of the standard Borel space X_{0} ×(2^{ℕ})^{ℕ} ×(ℂ^{n})^{ℕ×ℕ} ×(Y_{0}^{n})^{ℕ×ℕ} and so we endow it with the relative Borel structure. To see it is a standard Borel space we only need check that it is a Borel subset, and the first issue here is whether 7.6 (vi) corresponds to a Borel condition; this in turn follows using the precompactness of {a_{l} : l ∈ A_{j}}, as in the proof of 7.2. We also need to be concerned with showing 7.6 (ix) is Borel; but this amounts to the assertion that there are finite sequences i_{1}, i_{2}, ⋯, i_{n}, j_{1}, ⋯ j_{n} in ℕ and δ_{1}, ⋯, δ_{n} ∈ ℚ^{+}, δ′_{1}, ⋯, δ′_{n} ∈ ℚ^{+} such that

(a) each δ_{l} < δ′_{l},

(b) for all i′ ∈ ℕ and l ≤ n, if y_{i_{l},i′} < δ′_{l} (i.e., a_{i′} ∈ B_{δ′_{l}}(a_{i_{l}})), then i′ ∈ A_{j_{l}} (i.e. a_{i′} ∈ U_{j_{l}}); and so, by δ_{l} < δ′_{l}, we have B_{δ_{l}}(a_{i_{l}}) ⊆ U_{j_{l}};

(c) for all i′ with y_{i′,i} < 1 (i.e. a_{i′} ∈ B_{1}(a_{i})), we have some l with y_{i′,i_{l}} < δ_{l} (i.e. a_{i′} ∈ B_{δ_{l}}(a_{i_{l}})).

Granting all this, the projection function

((y_{i,i′}), (A_{i}), (ξ_{i,i′}), V_{i,j}, (φ_{i,j})) ↦ ((y_{i,i′}), (A_{i}), (ξ_{i,i′}))
M_{0}^{n} → X_{0} × (2^{ℕ})^{ℕ} × (ℂ^{n})^{ℕ×ℕ}

is one-to-one, Borel, and has M^{n} as its image. Thus, by Kechris [95, 15.1], M^{n} is a Borel subset of X_{0} × (2^{ℕ})^{ℕ} × (ℂ^{n})^{ℕ×ℕ} and therefore standard Borel.

⊣

It is easily seen that for each M, a complex manifold of dimension n, there is some p ∈ M^{n} with M_{p} and M isomorphic. The only restriction on our charts is that the sets V_{j} from 7.6 (vi) are basic open and connected, but refining a chart as needed we may always make this assumption.

Recall that for p, q ∈ M^{n} we set p ≅_{n} q if M_{p} and M_{q} are biholomorphic. We are finally in a position to verify the lemmas in §3-§6, whose proofs we postponed until now.

Proposition 3.3. ≅_{n} is Σ_{1}^{1}.

Proof. Fix p = ((y_{i,i′}), (A_{i}), (ξ_{i,i′})) and q = ((y′_{i,i′}), (A′_{i}), (ξ′_{i,i′})) in M^{n}, let V_{i}, U_{i} and V′_{i}, U′_{i} be the respective sets from 7.6 (vi) for p and q, and let φ_{i} and φ′_{i} be the respective functions. Note that we may parametrize the holomorphic functions from M_{p} to M_{q} by sequences of 4-tuples (V_{i,j}, W_{i,j}, ψ_{i,j}, k(i, j)) with (V_{ij}, W_{ij}, ψ_{ij}) in Y_{0}^{n} and k(i, j) ∈ ℕ, such that

(i) V_{i} is the union of {V_{i,j} : j ∈ ℕ};

(ii) W_{i,j} ⊆ V′_{k(i,j)};

(iii) for l ∈ A_{i} ∩ A_{i^{∗}}, φ_{i}(a_{l}) =_{df} ξ_{i,l}, φ_{i^{∗}}(a_{l}) =_{df} ξ_{i^{∗},l}, ξ_{i,l} ∈ V_{i,j}, ξ_{i^{∗},l} ∈ V_{i^{∗},j^{∗}}, we have

(φ′_{k(i,j)})^{−1}(ψ_{i,j}(ξ_{i,l})) = (φ′_{k(i^{∗},j^{∗})})^{−1}(ψ_{i^{∗},j^{∗}}(ξ_{i^{∗},l})).

(ii) and (iii) are clearly Borel; local compactness of the space gives the same conclusion for (i). The significance of (iii) is to ensure that the partial functions knit together in a well defined fashion and yield a holomorphic function from M_{p} to M_{q}. So given (V_{i,j}, W_{i,j}, ψ_{i,j}, k(i, j)))_{i,j∈ℕ×ℕ} as above we define

ψ : M_{p} → M_{q}

extending the assignment

ψ(a_{l}) = (φ′_{k(i,j)})^{−1}(ψ_{i,j}(ξ_{i,l}))

for l ∈ V_{i,j}.

Thus we have a Borel subset, Z_{0}^{n}, of M^{n} ×M^{n} ×(Y_{0}^{n} ×ℕ)^{ℕ×ℕ} that provides parameters for the collection of all pairs of complex n-dimensional manifolds and holomorphic functions between those spaces. For (p, q, u⃗) ∈ Z_{0}^{n}, let φu⃗ : M_{p} → M_{q} be the resulting holomorphic function. Since it is easily seen that

Z_{1}^{n} = {(p, q, u⃗, v⃗) : (p, q, u⃗), (q, p, v⃗) ∈ Z_{0}^{n}, φu⃗ = (φv⃗)^{−1}}

is a Borel subset of M^{n} × M^{n} × (Y_{0}^{n} × ℕ)^{ℕ×ℕ} × (Y_{0}^{n} × ℕ)^{ℕ×ℕ}, we then have that ≅_{n}, being the projection of Z_{1}^{n} (over u⃗, v⃗), is Σ_{1}^{1}. ⊣

Definition 7.8. We put D = {O ∈ O(ℂ) : O is connected}. It is not hard to see that D is a Borel subset of O(ℂ). For d ∈ D, we let D_{d} = d.

Recall that R is just M^{1}, the space of parameters of Riemann surfaces.

Proposition 3.5. There is a Borel function f : D → R such that for each d ∈ D, D_{d} ≅ M_{f(d)}.

Proof. For d = U an open subset of ℂ, we can let (a_{i}) enumerate U ∩ (ℚ + iℚ) and let (V_{i}) enumerate the basic open sets of diameter < 1 included in U. We may then let y_{i,i′} = d(a_{i}, a_{i′}), A_{j} = {i : a_{i} ∈ V_{j}} and ξ_{j,i} = a_{i} for a_{i} ∈ V_{j}. By local compactness of the spaces this can be done in a Borel in U fashion. (i)-(ix) of 7.6 then follow easily. ⊣

Lemma 4.5. Let M̂_{1} = ℂ_{∞}, M̂_{2} = ℂ, M̂_{3} = ℍ. For each i = 1, 2, 3, let T_{i} ⊆ R × F(M̂_{i}^{2}) be the set of (p, F) for which there exists

π : M̂_{i} → M_{p}

with π a covering map and F = F_{π} = {(x, y) : π(x) = π(y)}. Then T is Σ_{1}^{1}.

Proof. Following the proof of 3.1, we can obtain a standard Borel space Z_{2,i} of pairs (u⃗, p) such that u⃗ appropriately represents a holomorphic πu⃗ : M̂_{i} → M_{p}. The construction of 3.1 gives that for each ζ, ζ′ ∈ D, the fixed countable subset of M̂_{i} and q, r ∈ ℚ^{+} the set of pairs (u⃗, p) ∈ Z_{2,i} with d(πu⃗(ζ), πu⃗(ζ′)) < q and d(πu⃗(ζ), a_{n}) < r is Borel, where d is the metric for M_{p} and (a_{n}) the fixed dense subset of M_{p}.

Now the statement that πu⃗ be a covering map amounts to the requirement that each unit ball around each a_{n} in M_{p} can be covered by finitely many basic open sets, B_{δ_{1}}(a_{n_{1}}), ⋯, B_{δ_{k}}(a_{n_{k}}), so that for all j ≤ k we have, letting

W_{j} = π−1 ; u⃗(B_{δ_{j}}(a_{n_{j}})):

(i) any connected component of W_{j} is precompact and any two distinct connected components of W_{j} have disjoint closures;

(ii) for any connected component U of W_{j}, πu⃗|U : U → B_{δ_{j}}(a_{n_{j}}) is biholomorphic.

We fix j, and we verify that (i) and (ii) are Borel conditions. Put D_{j} = D ∩ W_{j}. Then if we define the equivalence relation:

ζ ∼ ζ′ ⇔ ζ, ζ′are in the same connected component of W_{j}, for ζ, ζ′ ∈ D_{j}, we see that the connected components of W_{j} are exactly the sets of the form

U_{C} = the interior of the closure of C,

where C is a ∼-equivalence class.

Thus (i), (ii) above are equivalent to (i)′, (ii)′ below, where C varies over all the ∼-equivalence classes:

(i)′ Every U_{C} is precompact and for any two distinct C_{1}, C_{2}, Ū_{C_{1}} ∩ Ū_{C_{2}} = ∅

(ii)′ πu⃗|U_{C} is biholomorphic.

By the methods of 7.2 and the proof of 3.1 we see that these conditions are Borel and thus the set of (u⃗, p) in Z_{2,i} such that πu⃗ is a covering map of M_{p} is Borel.

It is left to show that F = F_{πu⃗} can be expressed in a Borel manner. However, this amounts only to requiring that for each pair of precompact basic open U_{0}, U_{1} ⊆ M̂_{i}, we have that F ∩(Ū_{0}×Ū_{i}) ≠ ∅ iff there exists V_{0}, V_{1} basic open with V̄_{0} ⊆ U_{0}, V̄_{1} ⊆ U_{1} and ∀q ∈ ℚ^{+}∃ζ_{0} ∈ D ∩ V_{0}∃ζ_{1} ∈ D ∩ V_{1}(d(πu⃗(ζ_{0}), πu⃗(ζ_{1})) < q).

⊣

Proposition 4.6. There is a Borel function

g : ℍ^{ℕ} → D,

so that whenever x = (x_{n}) ∈ ℍ^{ℕ} enumerates a discrete set, we have that D_{g(x)} = g(x) = ℍ \ {x_{n} : n ∈ ℕ}.

Proof. First let A be the set of sequences (x_{n}) ∈ ℍ^{ℕ} enumerating a discrete subset of ℍ. This is Borel since (x_{n}) ∈ B if and only if

∀n∃δ ∈ ℚ^{+}∀m(m ≠ n ⇒ d(x_{n}, x_{m}) > δ).

For x ∉ A, we can just let g(x) = ℍ. For x ∈ A, we let g(x) = ℍ \ {x_{n} : n ∈ ℕ}. This is a Borel function, since for any basic open U ⊂ ℂ we have that U ⊆ g(x) if and only U ⊆ ℍ and no x_{n} is in U. ⊣

Recall that S_{d} is the space of discrete subgroups of P SL_{2}(ℝ). This is a Borel subset of F(P SL_{2}(ℝ)) in the Effros Borel structure, and hence a standard Borel space.

Lemma 5.3. There is a Borel map φ : S_{d} → R such that ℍ/G is conformally equivalent to M_{φ(G)} for all G ∈ S_{d} acting freely on ℍ.

Proof. Note that we can indeed verify in a Borel manner whether G ∈ S_{d} acts freely on ℍ: This amounts to the claim that for all basic open U ⊆ ℍ we may find a finite sequence V_{0}, V_{1}, ⋯, V_{n} of basic open sets covering Ū and such that for all i ≤ n,

(*) for all g ∈ G with g ≠ 1_{G}, g · V_{i} ∩ V_{i} = ∅.

However (*) is Borel, since it amounts to the assertion that for all W ⊆ P SL_{2}(ℝ) basic open not containing the identity, if G ∩ W ≠ ∅ then there exists h ∈ W such that h · V_{i} ∩ V_{i} = ∅.

So let us just fix G ∈ S_{d} acting freely on ℍ and describe φ(G).

First we let (V_{i}) enumerate the basic open sets which are G-discrete, in the sense of meeting each G-orbit in at most one point, and have diameter < 1 in the hyperbolic metric ρ (see 4.B). As in the proof of 3.3, this can be used to give a chart for a representative of ℍ/G in R. The one further problem is in uniformly obtaining a metric.

For any ζ, ξ ∈ ℍ and g ∈ G we have

inf_{h∈G}ρ(h · ζ, ξ) = inf_{h∈G}ρ(h · ζ, g · ξ),

and moreover this quantity is greater than zero if and only if G · ζ ≠ G · ξ. In particular

inf_{h∈G}ρ(h · ζ, ξ) = inf_{h,g∈G}ρ(h · ζ, g · ξ).

Therefore

d̃(G · ζ, G · ξ) = inf_{h,g∈G}ρ(h · ζ, g · ξ)

provides the needed metric on ℍ/G.

Thus if we let (a_{i}) enumerate a maximal G-discrete subset of ℍ ∩ (ℚ + iℚ) (in the sense that G · a_{i} ∩ G · a_{j} = ∅ for all i ≠ j), y_{i,i′} = d̃(G · a_{i}, G · a′_{i}), A_{i} = {j : G · a_{j} ∩ V_{i} ≠ ∅}, and ζ_{i,j} to be the unique element in V_{i} ∩ G · a_{j}, if it exists, we obtain from p = ((y_{i,i′}), (A_{i}), (ζ_{i,i′})) an element in R^{n} with M_{p} conformally equivalent to ℍ/G.

There is the further concern that all these steps can be performed in the Borel context, but this is routine and resembles earlier calculations. ⊣

Recall that for x ∈ ℝ^{ℕ} we let M(x) be the complex manifold

(𝔻 × ℂ) \ ⋃_{n∈ℕ}{1/(n + 2)} × {x_{n} + (n + 2), x_{n} + (n + 3), x_{n} + 2(n + 2)}

endowed with the inherited complex structure.

Lemma 6.3. There is Borel map

F : ℝ^{ℕ} → M^{2}

such that M(x) and M_{F (x)} are biholomorphic for all x ∈ ℝ^{ℕ}.

Proof. This follows the method of 3.3. ⊣

References

Beardon, A.F. [84] A Primer on Riemann Surfaces, London Math. Soc. Lecture Note Series, 78, Cambridge Univ. Press, 1984.

Becker, H. and A.S. Kechris [96] The Descriptive Set Theory of Polish Group Actions, London Math Soc. Lecture Note Series, 232, Cambridge Univ. Press, 1996.

Becker, J., C.W. Henson and L.A. Rubel [80] First-order Conformal Invariants, Ann. of Math, 112 (1980), 123-178.

Bedford, T., M. Keane, and C. Series (Eds.) [91] Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford Univ. Press, 1991.

Dellacherie, C. and P.-A. Meyer [83] Théorie Discrète du Potentiel, Hermann, Paris, 1983.

Dougherty, R., S. Jackson, and A.S. Kechris [94] The Structure of Hyperfinite Borel Equivalence Relations, Trans. Amer. Math. Soc., 341 (1) (1994), 193-225.

Forster, O. [81] Lectures on Riemann Surfaces, Springer-Verlag, New York, 1987.

Halmos, P.R., J. von Neumann [42] Operator Methods in Classical Mechanics, II, Ann. of Math., 43 (1942), 332-350.

Harrington, L.A., A.S. Kechris, and A. Louveau [90] A Glimm-Effros Dichotomy for Borel Equivalence Relations, J. Amer. Math. Soc., 3 (1990), 903-928.

Hjorth, G. [9?] Actions of the Classical Banach Spaces, preprint. Hjorth, G. [9?a] Classification and Orbit Equivalence Relations, preprint. Hjorth, G. and A.S. Kechris [95] Analytic Equivalence Relations and Ulm-

type Classifications, J. Symb. Logic, 60 (1995), 1273-1300.

Hjorth, G., A.S. Kechris, and A. Louveau [98] Borel Equivalence Relations Induced by Actions of the Symmetric Group, Ann. Pure and Appl. Logic, 92 (1998), 63-112. .

Imayoshi, Y. and M. Taniguchi [92] An Introduction to Teichmüller Spaces, Springer-Verlag, 1992.

Jackson, S., A.S. Kechris, and A. Louveau [9?] Countable Borel Equivalence Relations, in preparation.

Kaplansky, I. [69] Infinite Abelian Groups, Revised Ed., Univ. of Michigan Press, Ann Arbor, 1969.

Katok, S. [92] Fuchsian Groups, Univ. of Chicago Press, 1992. Kechris, A.S. [91] Amenable Equivalence Relations and Turing Degrees, J.

Symb. Logic 56 (1991), 182-194.

Kechris, A.S. [92] Countable Sections for Locally Compact Group Actions, Erg. Thy. and Dyn. Syst., 12 (1992), 283-295.

Kechris, A.S. [95] Classical Descriptive Set Theory, Springer-Verlag, 1995. Kechris, A.S. [98] Actions of Polish Groups and Classification Problems, to

appear in Analysis and Logic, Cambridge Univ. Press, 1998.

Kodaira, K. [86] Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, New York, 1986.

Patterson, A.T. [88] Amenability, Math Surveys and Monographs, 29, Amer. Math Society.

Rudin, W. [66] Real and Complex Analysis, McGraw-Hill, New York, 1966. Shelah, S. [84] Can You Take Solovay’s Inaccessible Away, Israel J. Math,

48 (1984), 1-47.

Thomas, S., and B. Velickovic [98], On the complexity of the isomorphism relation for finitely generated groups, preprint.

Wagon, S. [93] The Banach-Tarski Paradox, Cambridge Univ. Press, 1993.

Department of Mathematics, UCLA, Los Angeles, CA 90095; greg@math.ucla.edu Department of Mathematics, Caltech, Pasadena, CA 91125; kechris@caltech.edu