Countable Borel equivalence relations

Filippo Calderoni

These notes cover an introduction to countable Borel equivalence relations. The theory of countable Borel equivalence relation is a huge subject. It has a broad impact in several areas of mathematics like ergodic theory, operator algebras, and group theory, and has raised many chal- lenging open problems. We focus on some results about the (anti)classification of torsion-free abelian groups, some applications of Popa’s cocycle superrigidity theorem, and Martin’s conjec- ture. These notes are from a class I taught in Spring 2020. They may include typos and inaccuracies. Thanks to Will Adkins, John Baldwin, Paolo Degiorgi, Ozkan Demir, Yanlong Hao, Yexin Lin, Noah Schoem, and Kevin Zhou for helpful comments.

Contents

1 Background 2

2 Smoothness 7

3 Structural results 14

4 Hyperfiniteness 17

5 Treeability 24

6 Torsion-free abelian group of finite rank 27

7 Popa’s cocycle superrigidity 31

8 (Anti)classification of torsion-free abelian groups of finite rank 36

9 The algebraic structure of bireducibility degrees 41

1 10 Martin’s conjecture 45

A Reductions and absoluteness 53

Concept index 54

References 56

1 Background

Polish spaces

In this section we introduce the central notions of Polish and standard Borel spaces, and we list some fundamental theorems about them. For an extensive exposition on this topics and other fundamental aspects of descriptive set theory we refer the interested reader to Kechris’ book [23] or the excellent lecture notes of Tserunyan [46].

Definition 1.1. A topological space (X, τ) is said to be Polish if it is separable and metrizable (i.e., there is a complete metric that generates τ).

Example 1.2. Among examples of Polish spaces we have the following well-known topological spaces:

• For all n ∈ N, n = {0, . . . , n − 1} with the discrete topology is Polish. So is N and any countable set.

• Rn, Cn and the n-dimensional torus Tn, the unit interval I = [0, 1] and In, for all n ∈ N.

• All separable Banach spaces, such as the `p spaces for 1 ≤ p < ∞, and c0, the space of converging to 0 sequences with the sup norm.

• The Cantor space 2N, viewed as the product of infinitely many copy of {0, 1} with the discrete topology, is a Polish spaces. So is the Baire space NN. In both cases a compatible metric is given by d(x, y) = 2−(n+1) where n is the least integer such that x(n) 6= y(n).

We list a few useful facts abut Polish spaces.

Fact 1.3. Countable product of Polish spaces is Polish with respect to the product topology.

In particular, RN and [0, 1]N are all Polish spaces.

2 Fact 1.4. A subset of a Polish space is Polish (with respect to relative topology) if and only if it is Gδ (i.e., countable intersection of open sets).

Notice that Fact 1.4 applies to closed subset, since all closed subsets of any metric space are Gδ.

Theorem 1.5. Every Polish space is homeomorphic to a closed subset of RN.

Thus, the Polish spaces are, up to homeomorphism, exactly the closed subsets of RN.

Standard Borel spaces

Recall that a Borel (or measurable) space is a pair (X, B), where X is a set and B is a σ-algebra on X (i.e., a collection of subsets of X containing ∅ and closed under complements and countable unions). We refer to the sets in B as Borel sets. For any topological space (X, τ) we denote by B(τ) the σ-algebra generated by τ, which is defined as the smallest subset of P(X) containing τ. A common practice throughout this course, when we look at Polish spaces, is to “forget” the topology and consider the induced Borel structure.

Definition 1.6. A Borel space (X, B) is said to be standard if there exists a Polish topology τ on X that generates B as its Borel σ-algebra; in which case, we write B = B(τ). In this case, we call τ a compatible Polish topology.

An obvious example of a standard Borel space is (X, B(τ)), whenever (X, τ) is a Polish space. If (X, B) is a standard Borel space and Y ⊆ X is Borel, then we can regard Y as a Borel space with the relative Borel structure B  Y = {B ∩ Y : B ∈ B} = {B ∈ B : B ⊆ Y}. It is known that given a Polish space (X, τ) and A ∈ B(τ), there is a Polish topology τA ⊇ τ such that A is a clopen set in τA. (E.g., see Kechris [23, Theorem 13.1].) It follows that the the class of standard Borel spaces is closed under Borel subspaces:

Fact 1.7. If (X, B) is a standard Borel space and Y ∈ B, then (Y, B  Y) is also a standard Borel space.

If X is a topological space, denote by F(X) the set of its closed subset. We endow F(X) with the σ-algebra F generated by the sets

{F ∈ F(X) : F ∩ U 6= ∅}, where U is an open set of X. The measurable space (F(X), F) is called the Effros Borel space of X.

Fact 1.8. The Effros Borel space of any Polish topological space is standard.

3 The Effros Borel spaces have a lot of useful application. For example it can be used to define the (hyper)space of all Polish spaces and the (hyper)space of all separable Banach spaces.

Example 1.9 (The standard Borel space of all Polish spaces). By Theorem 1.5 every Polish space is (isomorphic to) an element of F(RN).

Example 1.10 (The standard Borel space of all separable Banach spaces). Any separable Banach space is linearly isometric to a closed subspace of C([0, 1]), the space of continuous functions

[0, 1] → R equipped with the sup norm. Then we define XBan as the subspace of F(C([0, 1])) given by the Borel subset of all closed linear subspaces of C([0, 1]) with the relative Borel structure. Each separable Banach space is isomorphic to an element of XBan.

Let (X, A) and (Y, B) be Borel spaces. A map f : X → Y is said to be Borel if f −1(B) ∈ A for each B ∈ B.A Borel isomorphism between X, Y is a bijection f : X → Y such that both f , f −1 are Borel. The following is a consequence of a deep result in descriptive set theory known as Souslin’s theorem.

Theorem 1.11. If X, Y are standard Borel spaces and f : X → Y, then the following are equivalent:

(1) f is Borel;

(2) Graph( f ) ⊆ X × Y is a Borel set.

Consequently, we obtain the following:

Theorem 1.12 (Luzin-Suslin). Let X, Y be Polish spaces and f : X → Y be Borel. If A ⊆ X is Borel and f  A is injective, then f (A) is Borel.

Next theorem shows that the theory of the Borel structure of Poish spaces is very robust.

Theorem 1.13 (Kuratowski). Any two uncountable Polish spaces are Borel isomorphic.

Thus, standard Borel spaces are fully classified by their cardinality, which can be either count- able or 2ℵ0 .

Luzin-Novikov uniformization theorem

A very common misconception about Borel sets is thinking that the image of any Borel set through a Borel function is itself Borel. This is true when the map is assumed to be injective (cf. Theo- rem 1.12) but it is false in general. For example the projection of a Borel measurable set in the

4 Y R

Rx

R∗

projX(R) X

Figure 1: Uniformization plane is not necessarily Borel measurable1. In this section we discuss other sufficient conditions for having Borel projection. This is tightly connected with the notion of uniformization and a classical uniformization theorem.

Definition 1.14. Given two standard Borel spaces X, Y and R ⊆ X × Y, a uniformization of R is ∗ ∗ a subset R ⊆ R such that R meets each vertical section Rx := {y ∈ Y : (x, y) ∈ R} in exactly ∗ one point. (See Figure 1.) Equivalently; R = Graph(g) with dom g = projX(R) and such that f (x) ∈ Rx, for all x ∈ dom(g). In this case, g is called uniformizing function.

If R ⊆ X × Y are as above, it follows from the axiom of choice that a uniformization for R always exists. It is natural to ask whether R ⊆ X, Y admits a “nice” uniformization. In particular, we are interested in the case when R is Borel and ask whether a Borel uniformization R∗ exists. ∗ Notice that a positive answer to this question will imply that projX(R) = projX(R ) is Borel subset of X by Luzin-Suslin theorem.

Clearly, a uniformization for R can be seen as the graph of a partial function g : dom(g) ⊆ X → Y. More in general, for topological spaces X, Y, a set A ⊆ X × Y is said a function graph if A = Graph(g) for some partial function g : X → Y; equivalently, for every x ∈ X, the vertical section Ax := {y ∈ Y : (x, y) ∈ R} has at most one element.

1This observation is attributed to Suslin, who noticed a mistake in a paper by Lebesgue asserting the contrary. More historical remarks can be found in Moschovakis [34]

5 Theorem 1.15 (Luzin-Novikov). Let X, Y be Polish spaces and R ⊆ X × Y be a Borel set all of whose sections Rx are countable. Then R has a Borel uniformization. F Moreover, R can be partitioned into countably many Borel function graphs R = n Graph(gn).

As an application of Theorem 1.15 we get that the class of Polish spaces is closed under countable-to-one Borel images:

Exercise 1.16. Let X, Y be Polish spaces, and f : X → Y be a countable-to-one Borel function. If A ⊆ X a Borel set, then f (A) is Borel.

We continue with a striking application of Luzin-Novikov’s theorem, which is central in the theory of countable Borel equivalence relations. Given a standard Borel space X, an equivalence relation E on X is said to be Borel if E is a Borel subset of X × X. We call E countable if every E-class is countable. In a similar way we call E finite if every E-class is finite. If G is a countable discrete group, then a standard Borel G-space is a standard Borel space X equipped with a Borel G-action (g, x) 7→ g · x. The induced orbit equivalence relation, denoted by X EG is the equivalence relation on X whose classes are the obits of the action.

Theorem 1.17 (Feldman-Moore [11]). If E is a countable Borel equivalence relation on the standard Borel X space X, then there exists a countable group G and a Borel action of G on X such that E = EG .

Proof. Since E is countable, E is a subset of X × X with countable sections. It follows from Theo- F rem 1.15 that E can be partitioned into the union of countably many Borel function graph n gn. Now consider E−1 := {(x, y) : (y, x) ∈ E}. By applying Luzin-Novikov again we obtain −1 F −1 −1 E = E = m hm. Then E = tm,n(gn ∩ hm ). So by possibly replacing (gn)n with (gn ∩ hm )n,m, we may assume without loss of generality that each gn is injective. We may assume without loss of generality that each gn is injective.

Now we want to extend each gn to a Borel involution of the whole space. We can do that if dom(gn) and ran(gn) are disjoint for all n. Since X is Hausdorff, the diagonal ∆(X) is closed. Then 2 S by second countability we can write X \ ∆(X) = (Un × Vn) where Un, Vn ⊆ X are disjoint open sets. It follows that

[ [ E = ∆(X) ∪ (E ∩ Um × Vm) = ∆(X) ∪ ( (gn ∩ (Um × Vm)) m m,n

6 Then we can extend each gn to an involution g˜n : X → X by setting  g (x) x ∈ dom(g ),  n n  g˜ (x) = −1 n gn (x) x ∈ ran(gn),   x otherwise.

S We have E = Graph(gn).

Let G = hg˜n | n ∈ Ni. It follows that

x E y ⇐⇒ ∃n(g˜n(x) = y) ⇐⇒ x EG y, as desired.

Remark 1.18. In fact, every countable Borel equivalence relation E on a Polish space X is a count- S able union of graphs of Borel involutions on X, i.e. E = n∈N Graph(gn), where each gn : X → X is a Borel involution. In particular, E is the orbit equivalence relation of a Borel action of a count- able group generated by involutions.

In this course we are interested in Borel equivalence relation. However, we can adopt a de- scriptive theoretical approach to analyze other sort of binary relations definable on standard Borel spaces. An interesting case is the one of quasi-orders, i.e., reflexive and transitive binary relations. The systematic study of definable quasi-order began with the fundamental paper of Louveau and

Rosendal [30]. A quasi-order Q on X is said to be countable if for all x0 ∈ X the set {x ∈ X : x Q x0} is countable. The following result, which describe countable Borel quasi-order from a dynamical point of view, is the analogue of Feldman-Moore theorem.

Theorem 1.19 (Williams [49]). If Q is a countable Borel quasi-order on the Polish space X, then there is a monoid M acting X in a Borel way such that x Q y ⇐⇒ ∃m ∈ M (x = m · y).

Proof. Exercise.

2 Smoothness

Definition 2.1. We say that a Borel equivalence relation E ⊆ X × X is smooth if there exists a Polish space Y, and a Borel function f : X → Y such that x E y holds if and only if f (x) = f (y) for all x, y ∈ Y.

7 An equivalent condition for smoothness is the existence a Polish space and a Borel injection

ι : X/E → Y that admits a Borel lifting (i.e., a Borel map f : X → Y such that f (x) = ι([x]E)).

Clearly, for any standard Borel space X, the equality relation =X is smooth. Smooth equivalence relations admit elements of some Polish space as complete invariants. Here are some natural examples.

Example 2.2. (1) Isomorphism on countable divisible abelian groups. The isomorphism relation on the standard Borel space of countable divisible abelian groups is smooth. To see this, recall that if A is a countable divisible abelian group, then A = D ⊕ T , where T is the torsion

subgroup and D is torsion-free. Let r0(A) ∈ N ∪ {∞} be the rank of D; and for each prime

p, let rp(A) ∈ N ∪ {∞} be the rank of the p-component Tp of T. Then the invariant

N ρ(A) = (r0(A), r2(A), r3(A),..., rp(A),... ) ∈ (N ∪ {∞})

determines A up to isomorphism.

(2) Isomorphism of Bernoulli shifts. Let (X, µ) be a probability space (X can be finite) and let µZ denote the product measure on XZ. Let S : XZ → XZ denote the shift automorphism; i.e., for f ∈ XZ and n ∈ N, S( f )(n) = f (n − 1). The dynamical system (XZ, µZ, S) is called a Bernoulli shift. By the measure isomorphism theorem, every Bernoulli shift is isomorphic to ([0, 1], λ, T), where λ is the Lebesgue measure and T some measure-preserving automorphism of ([0, 1], λ). In this case, we would call T a Bernoulli shift as well, and let B ⊆ Aut([0, 1], λ) be the set of all Bernoulli shifts. (Here Aut([0, 1], λ) is the Polish space space of all measure- preserving automorphisms of ([0, 1], λ) up to a.e. equality equipped with the weak topology.
Ornstein showed that B is a Borel subset of Aut([0, 1], λ), and hence is a standard Borel space. Furthermore, one can assign to each T ∈ B a real number e(T) called entropy. While the notion of entropy was introduced by Kolmogorov, Ornstein [37] also showed that e(T) com- ∼ pletely classify T up to isomorphism: that is T1 = T2 ⇐⇒ e(T1) = e(T2).

Now we see some useful characterizations of smoothness. A transversal for E is a set B ⊆ X which intersect each E-class in exactly one point. A selector for E is a function s : X → X such that s(x) E x and x E y implies s(x) = s(y).

Theorem 2.3. Let E be a countable Borel equivalence relation on X. The following are equivalent:

(i) E is smooth;

8 (ii) E admits a Borel transversal;

(iii) E admits a Borel selector.

Proof. (i) =⇒ (ii) Let Y be a Polish space and f : X → Y witness smoothness for E. Clearly f is a countable-to-one function, and its graph is a Borel set of X × Y with countable sections. It follows by Luzin-Novikov theorem that projY(Graph( f )) = f (X) is Borel (cf. Exercise 1.16), moreover there is a Borel uniformizing function g : f (X) → X. Then, the set g ◦ f (X) is a Borel transversal for E. (ii) =⇒ (iii) Let A ⊆ X be a Borel transversal for E. Define s : X → X by setting s(x) equal to the unique a ∈ A such that a E x. It is clear that s is a selector for E and Graph(s) = {(x, y) ∈ E : y ∈ A} is a Borel set, thus s is Borel by Theorem 1.11. (iii) =⇒ (i) A Borel selector s : X → X witnesses smoothness as x E y if and only if s(x) = s(y).

Remark 2.4. Note that the implication (iii) =⇒ (ii) is immediate because if s : X → X is a Borel selector for E, then we can define a Borel transversal by {x : s(x) = x}. On the other hand, the equivalence between (i) and (ii)–(iii) does not hold if we drop count- ability. In fact, there exists a Borel X ⊆ NN × NN without Borel uniformization and such N that proj1X = N and X (e.g., see [23, Exercise 18.17]). Next, we define E on X by declaring N x E y ⇐⇒ proj1(x) = proj1(y). Obviously, the map proj1 : X → N witnesses smoothness of E, but E does not admit a Borel selector, whose existence would be equivalent to Borel uniformization for X.

Exercise 2.5. Show that the action Z y R by n · x = x + n induces a smooth equivalence relation.

Exercise 2.6. Show that every finite Borel equivalence relation is smooth.

A countable Borel equivalence relation E is called aperiodic if every E-class is infinite.

Proposition 2.7. Suppose that E is an aperiodic countable smooth Borel equivalence relation on a standard

Borel space X. Then there is a sequence (Bn)n of pairwise disjoint Borel transversals of E such that X = S n Bn.

Proof. Let Y be a standard Borel space and f : X → Y be a Borel reduction from E to =Y. The set S Graph( f ) is Borel whose horizontal sections are countable. Then Graph( f ) = n gn for injective

9 S Borel partial functions gn : f (X) → X. Then An = ran gn is a Borel partial transversal for n An = X. → N ( ) = { ∈ N ∩ [ ] S } Define kn : X by setting kn x min k : Ak x E * m

A (countable) separating family for E is a sequence {An}n∈N of subsets of X such that

x E y ⇐⇒ ∀n(x ∈ An ⇔ y ∈ An).

Proposition 2.8. A countable Borel equivalence relation is smooth if it admits a countable separating family consisting of Borel sets.

Proof. Exercise.

We let X/E denote the equivalence classes with the quotient Borel structure, i.e. B ⊆ X/E is said to be Borel if and only if its inverse image in X is Borel.

Proposition 2.9. A countable Borel equivalence relation E is smooth if and only if the quotient X/E is a standard Borel space.

Proof. First assume that Z is a Polish space and f : X → Z witnesses smoothness. By countability of E, it follows that f is countable-to-one, so Y = f (X) is a Borel set by Exercise 1.16. Then Y is a standard Borel space with the induced Borel structure. Using the fact that f is countable-to-1 we can check that the bijection X/E → Y induced by f is a Borel isomorphism. It follows that X/E is standard.

Conversely, if the induced Borel structure on X/E is standard, then the map x 7→ [x]E witnesses that E is smooth.

Ergodicity

In this section we introduce ergodicity, a measure theoretical obstruction to smoothness. Let (X, B) be a Borel space. A (Borel) measure on X is a function µ : B → R+ ∪ {∞} that takes ∅ to 0 and that is σ-additive, i.e. for pairwise disjoint Borel sets An, for n ∈ N, we have   [ µ An = ∑ µ(An). n n

10 If µ is a measure on X and Y ⊆ X is a Borel set, then µ  Y is the measure on Y which is the restriction of µ to the Borel subsets of Y. Moreover, when f : X → Y is a Borel function between

Polish spaces and µ is a Borel measure on X, we can define the push-forward measure f∗µ on Y by −1 setting f∗µ(B) = µ( f (B)), for every Borel B ⊆ Y. We call a Borel measure µ on X finite if µ(X) < ∞. In case µ(X) = 1, we call µ a probability measure. Until further notice we assume that µ is a finite Borel measure. A set A ⊆ X is called µ-measurable if there is some Borel B ⊆ X so that µ(A ∆ B) = 0. In this case, we set µ(A) = µ(B).

Definition 2.10. A measure µ on X is said to be E-ergodic if for every measurable A ⊆ X, if A is E-invariant then µ(A) = 0 or µ(X r A) = 0. A measure µ is called non-atomic if every singleton has measure 0.

Proposition 2.11. Let E be a countable Borel equivalence relation on a standard Borel space. If X has an E-ergodic, nonatomic measure µ, then E is not smooth.

Proof. Suppose that E is smooth towards contradiction. Let {An}n∈N be a countable Borel gener- ating family for E. Each An is invariant, so either µ(An) = 0 or µ(X r An) = 0 by ergodicity. Then the set \ \ C = {An : µ(X r An) = 0} ∩ {X r An : µ(An) = 0}} has positive measure. However, C = [x]E for some x ∈ X. It follows that µ(C) = 0, since µ is non-atomic.

N Definition 2.12. For x, y ∈ 2 we define x E0 y ⇐⇒ ∃m ∀n ≥ m (x(n) = y(n)).

N 1
Fact 2.13. The product probability measure on 2 where each bit in {O, 1} has measure 2 is non-atomic and E0-ergodic. Thus E0 is not smooth.

Generic ergodicity

In this subsection we discuss another restriction to smoothness. Until further notice, let X be a Polish space.

Definition 2.14. A set A ⊆ X is nowhere dense if its closure has nonempty interior.

1 + Examples of nowhere dense sets include Z and { n : n ∈ N } in R.

Definition 2.15. A ⊆ X is meager if it is countable union of nowhere dense sets. We call A ⊆ X comeager if its complement is meager.

11 An example of meager set is Q in R. Likewise the sets of measure 0 with respect to a given measure, the set of all meager subsets of a topological space forms a σ-ideal. Hence we can define a notion of measurability with respect to meager sets.

Definition 2.16. A ⊆ X is Baire measurable if there is an open set U ⊆ X such that A∆U is meager. Moreover, we call a function f : X → Y Baire measurable if the preimage of every open set in Y is Baire measurable.

Fact 2.17. The set of Baire measurable subsets of X is the smallest σ-algebra containing the open sets and the meager sets.

In particular, all Borel sets are Baire measurable. The existence of subsets of R that are not Baire measurable is independent from ZF. Using the axiom of choice it is possible to build a so-called Bernstein set that is not Baire measurable. On the other hand, the Axiom of Determinacy (AD) implies that all subsets of a Polish space are Baire measurable.

Definition 2.18. An equivalence relation on X is called generically ergodic if every E-invariant Baire measurable set is either meager or comeager.

Proposition 2.19. Suppose that E is an equivalence relation on X and f : X → 2N is a Baire measurable homomorphism between E and ∆(2N) (i.e., x E y implies f (x) = f (y)). If E is generically ergodic, then N −1 there is x0 ∈ 2 such that f (x0) is comeager.

Proof. We define an increasing sequence (sn)n of finite binary sequences such that |sn| = n and −1 −1 f (Nsn ) is comeager. Put s0 = ∅. As inductive hypothesis we assume that f (Nsn ) is comeager. Notice that since f is a Baire measurable homomorphism, the preimage of every open set is E- −1 invariant and Baire measurable. So, f (Nt) is either meager or comeager by generic ergodicity. Since we have −1 −1 −1 f (Ns ) = f (N _ ) ∪ f (N _ ), n sn 0 sn 1

−1 _ S there is some i ∈ {0, 1} such that f (N _ ) is comeager. Set sn+ = sn i and put x0 = sn. It sn i 1 n follows that −1 −1 \ \ −1 f (x0) = f ( Nsn ) = f (Nsn ) n n which is a countable intersection of comeager sets, hence comeager.

Corollary 2.20. Let E be an equivalence relation on X. If E is generically ergodic and every equivalence class is meager, then E is not smooth.

12 Proof. Suppose that f : X → 2N is a Borel reduction. In particular, f is a Baire measurable reduc- tion, hence the preimage of every point is a single E-class, which is meager by the assumption. However, by Proposition 2.19, there is a point in 2N whose preimage is comeager.

Fact 2.21. Suppose G y X in a continuous fashion. The following are equivalent:

(1) EG is generically ergodic.

(2) Every invariant nonempty open set is dense.

(3) For comeager-many x ∈ X, the orbit G · x = [x] is dense.

(4) There is a dense orbit.

(5) For every nonempty open sets U, V ⊆ X, there is g ∈ G such that (gU) ∩ V 6= ∅.

Example 2.22. The Vitali equivalence relation is generated by the action Q y R by q · x = x + q.

Exercise 2.23. Show that E0 is not smooth using generic ergodicity.

Let G act on X by homeomorphism. A minimal invariant set for the action is a closed, G- invariant subset M ⊆ X such that: if C ⊆ X is any closed, G-invariant set and C ∩ M 6= ∅ then C = M.

Proposition 2.24. Suppose that G is a countable group acting by homeomorphisms on a compact Polish space X. Then there exist a minimal set M ⊆ X for the action of G.

Proof. Let S := {A ⊆ X | A is nonempty, closed and G-invariant}. We note that S is nonempty because X ∈ S and it is partially ordered under inclusion. A consequence of the compactness T of X is that every chain Aii∈I satisfies the finite intersection property. It follows that i∈I Ai is nonempty and necessarily closed and G-invariant. Therefore, every chain in X admits has lower bound. Then, using Zorn’s Lemma, there exists a minimal invariant set M ⊆ X for the action G y X.

Exercise 2.25. Let G y X continuously. If A ⊆ X is G-invariant, then so is A.

Exercise 2.26. Let ∅ 6= M ⊆ X. The following are equivalent:

(i) M is minimal;

(ii) For all x ∈ X, we have G · x = M.

13 Proposition 2.27 (See also Calderoni-Clay [6]). Suppose that G is a countable group acting by homeo- morphisms on a compact Polish space X such that EG is smooth. Then there exists a finite orbit.

X X Proof. Suppose that EG is smooth. Let M ⊆ X be a minimal set for the action of G on X. Since EG M is smooth, so is the action EG . Note that every G-orbit of every point of M is dense in M, so by

Proposition 2.24, M there must be x0 ∈ M such that G · x0 is nonmeager. S However, G · x0 = g∈G{g · x0} thus there is some g0 ∈ G such that {g0 · x0} is open in M. Since every is dense in M, it follows that M must in fact consist of a single orbit, precisely M = G · x0. But M ⊆ X is countable and closed. Thus M is a compact space with the relative topology, and every point in M is isolated because G acts continuously. This is not possible if M is infinite.

3 Structural results

Suppose that E and F are Borel equivalence relations on standard Borel spaces X and Y, respec- tively.

Definition 3.1. We say that E is Borel reducible to F (in symbols E ≤B F) if and only if there exists a Borel function f : X → Y such that x E y ⇐⇒ f (x) F f (y). Such function f is called Borel reduction. If moreover f is injective we say that F is a Borel embedding and E is Borel embeddable into F (written

E vB F).

Exercise 3.2. Suppose that E is a Borel equivalence relation on a standard Borel space X with infinitely many equivalence classes. Let x0 ∈ X. Then E is Borel reducible to E  (X r [x0]E).

Occasionally we will be using a weaker notion to compare equivalence relations.

Definition 3.3. A function h : X → Y is a Borel homomrphism between E and F if it is Borel and x E y =⇒ f (X) F f (y).

Theorem 3.4 (Silver’s dichotomy [39]). For every Borel (in fact coanalytic) equivalence relation on a standard Borel space, exactly one of the following holds:

(I) There are only countably many E-classes;

(II) idR vB E.

The original prove of Silver original is quite involved and uses a forcing argument. Another proof was recently found by Miller who deduced Silver’s dichotomy from a dichotomy result

14 of Kechris, Solecki, and Todorceviˇin Borel combinatorics on Borel colorings for analytic graphs [26]. Another alternative proof of Silver’s dichotomy was given by Harrigton, who used the so- called Gandy-Harrington topology and methods from recursion theory. Interestingly, the Gandy- Harrington topology and those effective methods are key in the proof of Harrington, Kechris, and Louveau of another remarkable dichotomy theorem on Borel equivalence relations.

Theorem 3.5 (Glimm-Effros dichotomy [16]). Let E be a Borel equivalence relation on a standard Borel space. Then either

(I) E is smooth; or

(II) E0 vB E.

The reason why Theorem 3.5 is named after Glimm and Effros is because it generalizes results of both from the ’60s. In fact, Glimm [15] proved a particular case of Theorem 3.5 for equivalence relations induced by continuous action of locally compact Polish groups. Glimm’s result was generalized afterwards by Effros [9] to Fσ orbit equivalence relation induced by continuous actions of Polish groups. Combining Theorem 3.4 with Theorem 3.5 it is clear that an initial segment of the class of countable Borel equivalence relation have the following linear structure

id(1)

Universality

At the other extreme of smoothness we encounter the phenomenon universality.

Definition 3.6. A countable Borel equivalence relation F is called universal if E ≤B F for every countable Borel equivalence relation E.

Given any standard Borel space X and a countable group G, denote by XG the set of functions from G to X with the product Borel structure. The shift action G on XG given by g · f (a) = f (g−1a) for f ∈ XG, g ∈ G.

Definition 3.7. We denote by E(G, X) the orbit equivalence relation generated by the shift action of G on XG.

15 Exercise 3.8. If G is homomorphic image of H, then E(G, X) ≤B E(H, X).

Exercise 3.9. If G is a subgroup of H, Then E(G, X) ≤B E(H, X).

Let F∞ be the countable non-abelian free group on infinitely many generators.

N F Theorem 3.10. E∞ = E(F∞, (2 ) 2 ) is a universal countable Borel equivalence relation.

X Proof. Let E = EG for some countable G y X. Since any countable group is homomorphic image of F∞, we can suppose that E is induced by some Borel action F∞ y X.

Since X is standard Borel there exists a sequence {Un}n∈N of Borel subsets of X which separates points. Define a map f : X → (2N)F2 by

−1 fx(a)(i) = 1 ⇐⇒ a · x ∈ Ui

It is immediate that f is injective, moreover x E y ⇐⇒ fx E∞ fx because

−1 g · fx(a)(i) = 1 ⇐⇒ fx(g a)(i) = 1

−1 ⇐⇒ (a g)x ∈ Ui

−1 ⇐⇒ a (gx) ∈ Ui

⇐⇒ fgx(a)(i) = 1.

The following Borel reductions were obtained by Dougherty, Jackson, and Kechris [7].

Z−{0} Fact 3.11. E(G, 2 ) ≤B E(G × Z, 3)

Fact 3.12. E(G, 3) ≤B F(G × Z2, 2).

Proposition 3.13. E(F2, 2) is a universal countable Borel equivalence relation.

n n −n Proof. Since F2 = ha, bi has a subgroup isomorphic to F∞ (for example take ha b a | n ≥ 1i), it suffices to show that E(F∞, 2) is universal. Then we have

N Z−{0} E(F∞, 2 ) ≤B E(F∞, 2 ) ≤B E(F∞ × Z, 3) ≤B E(F∞ × Z × Z2, 2) ≤B E(F∞, 2)

It follows that every group containing a nonabelian free subgroup free subgroup induces a universal countable Borel equivalence relation. Whether the converse is also true is still open.

Definition 3.14. A countable group G is said to action universal if there is some standard Borel X space X and some Borel action G y X so that EG .

Problem 3.15. Is it true that if G is action universal then G contains a nonabelian free subgroup?

Recent work of Thomas [45] suggests that the answer might be negative.

16 4 Hyperfiniteness

Definition 4.1. A Borel equivalence relation E is hyperfinite if there is an increasing sequence of S finite Borel equivalence relations F0 ⊆ F1 ⊆ F2 ⊆ · · · such that E = n Fn.

Notice that all hyperfinite equivalence relations are in fact countable.

Example 4.2. (1) E0 is hyperfinite. To see this define Fn by

x Fn y ⇐⇒ ∀m ≥ n (x(m) = y(m)).

S It is clear that E0 = n Fn.

(2) If G is a Polish group (i.e., a topological group whose topology is Polish) and H is a subgroup S of G that can be written as H = Hn such each Hn is a countable discrete subgroup of G and

Hn ⊆ Hn+1 for n ≥ 1. Then the coset equivalence relation on G, which is defined by

G g EH h ⇐⇒ h ∈ gH,

S 1 is hyperfinite. As an application we obtain that Ev is hyperfinite as Q = n n! Z.

Proposition 4.3. If E smooth, then E is hyperfinite.

X Proof. Let E = EG for some countable group G. Suppose that G = {g0, g1,...} with g0 = 1. Since E is smooth, E admits a Borel selector s : X → X by Proposition 2.3–(iii). Then define an sequence of finite Borel equivalence relations Fn, for n ≥ 1, by

 n n  _ _ x Fn y ⇐⇒ x = gis(x) and y = gis(x) or x = y. i=0 i=0

Then for any Fn, each class is either a singleton or has at most n + 1 elements. Hence Fn is a finite S Borel equivalence relation and E = n Fn.

The following closure properties are easy to verify.

Fact 4.4. (i) If E ⊆ F and F is hyperfinite, then E is hyperfinite.

(ii) If E is hyperfinite and Y ⊆ X is Borel, then E  Y is hyperfinite.

Given an equivalence relation E on X, a subset A ⊆ X is a complete section for E is it meets every E-class (i.e., for all x ∈ X, [x]E ∩ A 6= ∅).

17 Proposition 4.5. Let E be a countable Borel equivalence relation on X. If A ⊆ X is a Borel complete section for E such that E  A is hyperfinite, then E is hyperfinite.

Proof. Suppose that F0 ⊆ F1 ⊆ · · · are finite Borel equivalence relations on A such that E  A = S n Fn. Let E = EG and G = {g0, g1,... } such that g0 = 1. For each x ∈ X let n(x) be the least index such that gn(x)x ∈ A. Define Fen by

x Fen y ⇐⇒ n(x) ≤ n and n(y) ≤ n and gn(x)x Fn gn(y)y or x = y.

S Each Fen is a finite Borel equivalence relation. So E is hyperfinite as E = n Fen

Exercise 4.6. If F is hyperfinite and E ≤B F, then E is hyperfinite.

Proposition 4.7 ([7]). Let E be a countable Borel equivalence relation. Then the following are equivalent:

(i) E is hyperfinite.

S (ii) E is hypersmooth, i.e., E = n∈N Fn where (Fn)n∈N is an increasing sequence of smooth Borel equivalence relations.

(iii) There is a Borel action a : Z y X such that E = Ea

Proof. The implication (1) =⇒ (2) is clear since every finite Borel equivalence relation is smooth (see 2.6). For (2) =⇒ (1) we refer the reader to [7]. For (1) =⇒ (3), first suppose that E is smooth. Notice that X can be partitioned into F n∈N∪{∞} Xn with Xn = {x ∈ X : |[x]E| = n}. Clearly each Xn is E-invariant and E  Xn is smooth. We may assume that each E-class has the same cardinality. If each E-class has cardinality n ∈ N, let < be a Borel well-ordering of X. Then, define  + x if x has an immediate successor in <  [x]E T(x) =  min[x]E otherwise.

Clearly T generates a Borel Z-action by n · x = Tn(x) and this action induces E. On the other hand, if E is aperiodic, then there exists a sequence (Bn)n∈Z of pairwise disjoint of Borel transversals for S E so that X = n Bn (see Proposition 2.7.) For any x ∈ Bn, we define T(x) = y for the unique y ∈ X such that x E y and y ∈ Bn. It is clear that E is induced by the Z-action generated by T.

Now assume that E is hyperfinite. Let (Fn)n∈N be an increasing sequence (Fn)n ∈ N of finite S Borel equivalence relations on X such that F0 is equality and E = n Fn. Also let < be a Borel linear ordering on X.

18 Since each Fn is finite equivalence relation, it is smooth. So there exists a Borel selector sn : X →

X for each Fn. Given distinct x, y ∈ X, let n(x, y) denote the maximal natural number n for which sn(x) 6= sn(y), and put x ≺ y ⇐⇒ sn(x) < sn(y). Then ≺ is a Borel partial order on X whose restriction to any E-class is isomorphic to the usual ordering of N, −N, or Z. Notice that E is smooth on the union of the E-classes C for which ≺ C is not isomorphic to Z because we can use the existence of a maximal or a minimal element to define a Borel transversal. On the remaining classes we can define a Z-action generating E in the natural way. The proof of (3) =⇒ (1) relies on the so called Marker lemma.

Lemma 4.8. Every aperiodic countable Borel equivalence relation admits a vanishing sequence of markers, i.e., a sequence of Borel complete sections that intersect every E-class.

In each E-class C we define the Z-ordering

x 0 (n · x = y).

Let (An)n be a vanishing sequence of markers for E. Note that the union of the set of all classes

C in which An ∩ C has a least or largest element in

Then define Fn, for n ∈ N, by

x Fn y ⇐⇒ x = y or (x E y and [x, y] ∩ Bn = ∅),

S where [x, y] = {z ∈ [x]E | x < z < y}. We have E = n Fn.

Exercise 4.9 (See [51, Lemma 3]). Let G be a virtually infinite cyclic group, that is, a group with a finite index subgroup isomorphic to Z. Then any Borel action of G on a standard Borel space induces a Borel hyperfinite equivalence relation.

In view of 3.5 E0 is minimal among the nonsmooth hyperfinite Borel equivalence relations. It turns out that E0 is also a universal element for this class.

Theorem 4.10 ([7, Theorem 7.1]). If E is Borel hyperfinite, then E ≤B E0.

As a consequence, being reducible to E0 is equivalent to (1)–(3) of Theorem 4.7. An alternative proof of Theorem 4.10 was given by Hjorth (see [47]).

19 It follows from Theorem 4.10 that all nonsmooth hyperfinite equivalence relation are Borel bi- reducible. However, hyperfinite Borel equivalence relation admits have a finer classification up to Borel isomorphism.

Definition 4.11. Let E be a countable Borel equivalence relation on a standard Borel space X.A (probability) measure on X is said E-invariant if there is some countable G acting on X in a Borel fashion and E is invariant under this action. Denote by EINV(E) the set of E-ergodic E-invariant Probability measures on X.

ℵ For any countable Borel equivalence relation E, |EINV(E)| ∈ {0, 1, 2, . . . , n,..., ℵ0, 2 0 }. By results of Farrel [10] and Varadarajan [48] a countable Borel equivalence relation E that admits an E-invariant probability measure, also admits an E-ergodic one.

∼ Theorem 4.12 ([7]). Let E, F be aperiodic, non-smooth hyperfinite Borel equivalence relations. Then E =B F if and only if |EINV(E)| = |EINV(F)|.

For example E0 has a unique invariant, ergodic, probability measure, while the orbit relation F(Z, 2) induced by the shift action of Z on the free part of 2Z admits 2ℵ0 invariant ergodic proba- bility measures (consider product measures corresponding to the (p, 1 − p) measure on {0, 1} for 0 < p < 1).

The union problem

We pointed out that hyperfiniteness coincide with hypersmoothness. It is natural to ask whether hyper-hyperfinite implies hyperfinite. This is an interesting major open problem in this area.

Question 4.13 (Union problem). Let E1 ⊆ E2 ⊆ E3 ⊆ · · · be an increasing sequence of hyperfinite S equivalence relations. Is E = n En hyperfinite?

Theorem 4.14 (Dye [8]; Krieger [27]). Let X be a standard Borel space and µ a probability measure on X. If E is the union of an increasing sequence of hyperfinite Borel equivalence relations, then E is hyperfinite

µ-a.e., i.e., there is a conull Borel set A ⊆ X such that E  A is hyperfinite.

Unfortunately to attack this and other problem about hyperfiniteness we cannot use Baire category methods. In fact, every countable Borel equivalence relation is hyperfinite generically.

Theorem 4.15 (Hjorth-Kechris). Suppose that E is a countable Borel equivalence relation on a Polish space X. Then there is an E-invariant comeager Borel set C ⊆ X such that E  C is hyperfinite.

20 This is far from true in measure theoretical context. For example, consider the equivalence F relation E(F2, 2) on X = 2 2 with the usual product measure. For every conull Borel set B ⊆ X, the restriction E  A is not hyperfinite.

Amenable groups

Suppose that X is a set. A finitely additive probability measure (f.a.p.m.) on A is a map

µ : P(X) → [0, 1] such that µ(X) = 1 and whenever A, B ⊆ X are disjoint then µ(A ∪ B) = µ(A) + µ(B). If G is a countable group and µ is a f.a.p.m. on G we say that µ is left-invariant if µ(gA) = µ(A) for all g ∈ G and A ⊆ G.

Definition 4.16. A countable group is amenable if it admits a left-invariant finitely additive proba- bility measure.

Example 4.17. 1. Every finite group G is amenable with left-invariant f.a.p.m. µ(A) = |A|/|G|.

2. Z is amenable. In fact, if U is a nonatomic ultrafilter on N we can define a left-invariant |A∩{−n,...,n}| f.a.p.m. by µ(A) = limn∈U 2n+1 .

Fact 4.18. (i) If H is amenable and G is a subgroup of H, then G is amenable.

(ii) If N is a normal subgroup of G, G is amenable if and only if both N and G/N are amenable. In particular, amenable groups are closed under epimorphic images.

(iii) G is amenable if and only if every finitely generated subgroup is amenable.

If X, Y ⊆ X we write X ∼ Y if there are partitions X1,..., Xn and Y1,..., Yn of X, Y respectively and group elements g1,..., gn ∈ G such that giXi = Yi.

Definition 4.19. A group G is paradoxical if there are disjoint sets A, B ⊆ G such that A ∼ B ∼ G.

Proposition 4.20. If G is paradoxical, then G is not amenable.

Proof. Suppose that µ is a left-invariant finitely additive probability measure on G, and fix disjoint A, B ⊆ G such that A ∩ B = ∅ and A ∼ B ∼ G.

µ(G) ≥ µ(A ∪ B) = µ(A) + µ(B) = 2µ(G) which is a contradiction.

21 In fact, we have the following:

Theorem 4.21. G is not amenable if and only if G is paradoxical.

Proposition 4.22. The nonabelian free group on two generators F2 is paradoxical.

±1 ±1 Proof. Let F2 = ha, bi. For each x ∈ {a , b }, let S(x) be the set of reduced words which begin with x. −1 −1 −1 Let A = S(a) ∪ S(a ) and B = S(b) ∪ S(a ). Since aS(a ) = F2 \ S(a), it follows that

A ∼ F2. Similarly, we get B ∼ F2, thus F2 is paradoxical.

Another characterization of amenability uses measure theory. Let X be a separable and metrizable topological space. Denote by P(X) the set of all probability measure on X. We give P(X) the topology whose basis is formed by the sets Z Z

Uµ,e, f1,..., fn = {µ ∈ P(X) : | fidν − fidµ| < e} for any i = 1, . . . n, µ ∈ P(X), e > 0, fi bounded real valued function on X.

Fact 4.23 ([23, Section 17.E]). If X is Polish, then P(X) is Polish. If moreover X is compact, then P(X) is a compact Polish space.

Note any continuous action G y X induces a continuous G-action on P(X) by

gµ(A) = µ(g−1 A), for all µ measurable A ⊆ X.

Definition 4.24. We say that µ ∈ P(X) is G-invariant if gµ = µ for all g ∈ G.

Theorem 4.25. Let G be a countable group. Then G is amenable if and only if for every compact metric space K and every continuous action G y K there exists a G-invariant measure in P(K).

With this in mind we can prove the following:

Theorem 4.26. Suppose that G acts on X freely, and there is a G-invariant probability measure µ on X. If X EG is hyperfinite, then G is amenable.

X Y Suppose that G acts on X, and H acts on Y freely. If f is a Borel homomorphism from EG to EH then we can define a Borel map α : G × X → H by

α(g, x) = h ⇐⇒ h · f (x) = f (g · x).

22 Notice that, since the action of H on X is free, α is well-defined. Moreover it is straightforward that α satisfies α(gh, x) = α(g, hx)α(h, x) for all g, h ∈ G and x ∈ X. A map with such property is called Borel cocycle.

Definition 4.27. Suppose that α : G × X → H is a Borel cocycle and H acts on Y in a Borel way. A µ-measurable map f : X → Y is called α-invariant if

α(g, x) f (x) = f (gx)

µ-almost everywhere.

Theorem 4.28 (Zimmer; see also [20, B 3.1]). Let G be a countable amenable group acting on a standard Borel space X with invariant probability measure µ in a Borel fashion. Moreover let H be a countable group acting continuously on a compact metric space K. For any cocycle α : G × X → H, there exists an

α-invariant µ-measurable map from X → P(K), x 7→ νx.

Proof of Theorem 4.26. Let G act freely on X, and µ be a G-invariant probability measure. Since E is hyperfinite E = EZ for some Borel action of Z on X. The free action of G on X induces a cocycle α : Z × X → H by α(n, x) = g if and only if nx = gx. Now suppose that K is a compact metric space and G acts continuously on K. We claim that there exists a Borel invariant probability measure ν on X. Consider the continuous action of G on P(K) by gµ(A) = µ(g−1 A). By theorem 4.28 there exists an α-invariant µ-measurable map

X → P(K), x 7→ νx. That is, α(n, x)νx = νnx = νgx with g = α(n, x). For A ⊆ X Borel, define Z ν(A) = νx(A)dµ. X Clearly ν is a probability measure, so it is left to show it is invariant.

Z ν(gA) = νx(gA)dµ = X Z −1 = g νx(A)dµ = X Z = νg−1x(A)dµ. X

R However the integral in the last line equals X νx(A)dµ = ν(A). So ν is invariant as desired.

23 Corollary 4.29. The equivalence relation E(F2, 2) is not hyperfinite.

Proof. Let µ be the usual product measure 2F2 . Denote by (2)F2 the free part of the shift action F F F F F F2 y 2 2 . That is, (2) 2 = {x ∈ 2 2 : ∀g 6= 1 g · x 6= x}. Clearly, F2 acts on (2) 2 freely and (2) 2 is F a F2-invariant Borel set. Let F(F2, 2) be the equivalence relation induced by the action F2 y (2) 2 .

G F Claim. Let G be a countable group and µG is the product measure on 2 , then µG((2) 2 ) = 1.

Proof of the Claim. Exercise.

F Since µ(F2) = 1, the restriction of µ on (2) 2 is a probability measure. We know that F2 is not amenable. (See Proposition 4.22.) It follows by Theorem 4.26 that F(F2, 2) is not hyperfinite. Since hyperfinite equivalence relations are closed under restriction, E(F2, 2) is not hyperfinite either.

5 Treeability

Let E be a countable Borel equivalence relation on a standard Borel space X.A Borel graphing of E is a Borel graph G = (X, R) whose connected components are exactly the E-classes. A Borel treeing of E is a graphing of E whose connected components are acyclic.

Definition 5.1. A Borel equivalence relation E is treeable if it admits a Borel treeing.

Example 5.2. Hyperfinite Borel equivalence relations are treeable.

Proposition 5.3. Let Fn be the free group on n generators. Suppose that Fn acts freely on a standard Borel space X in a Borel fashion. Then EX is treeable. Fn

Proof. Let F be the free group with generators (g ) and let E = EX F . For each C ∈ X/E and n i i

(x, y) ∈ T ⇐⇒ ∃i (gi · x = y or x = gi · y).

A consequence of the previous proposition is that F(F2, 2) is treeable. Hence the class of hy- perfinite equivalence relations is strictly included in the treeable ones.

Proposition 5.4. Let E be a countable Borel equivalence relation on a a standard Borel space X.

(i) If E is treeable and A ⊆ X is Borel, then E  A is treeable.

(ii) If A ⊆ X is a complete Borel section E and E  A is treeable, so is E.

24 (iii) If E ≤B F and F is treeable, then so is E.

(iv) If E ⊆ F and F is treeable, so is E

Proof. (i) Let (E, T) be a treeing for E with connected components TD, for D ∈ X/E. Let G =

(gi)i∈N. For every x ∈ X, we consider the lexicographic order on the paths in T[x] starting at x defined as follows:

0 0 (x, x1,..., xm)

Now fix C an equivalence class of E  A and let D = [C]E. For all x ∈ D let ρ(x) be the end of −1 the

if ρ(x) = c and x, x1,... xm−1, c is the

(ii) Let f : X → A be a Borel map with f (x) E x, and f (x) = x for x ∈ A. For each C ∈ X/E, let

TA∩C be the associated tree. We can define a tree TC from TA∩C by adding an edge (x, f (x)) from each x ∈ X \ A. The proof of (iii) follows by the previous items, while (iv) can be proved as (i).

X Definition 5.5. Let X be a standard Borel space and E = EG be a countable Borel equivalence relation. A Borel cocycle for E is a Borel map ρ : E → G such that

x E y E z =⇒ ρ(x, z) = ρ(y, z)ρ(x, y)

Exercise 5.6. Suppose that ρ : E → G be a Borel cocycle of E. Then for all x, y ∈ X,

(a) ρ(x, x) = 1;

(b) if x E y then ρ(x, y) = ρ(y, x)−1.

Definition 5.7 ([19]). An action a : G y X has the cocycle property if there is a Borel cocycle ρ : E → G with ρ(x, y) · x = y.

25 If G acts on X freely then we can easily define Borel cocycle ρ : EG → G. For each x EG y we set ρ(x, y) equal to the unique g ∈ G such that g · x = y.

Theorem 5.8. For a countable Borel equivalence relation E the following are equivalent:

(1) E is treeable;

X (2) For all countable G and for all G-action such that E = EG , this action has the cocycle property.

X (3) For all countable G such that E = EG there is a free Borel action of G on some Borel Polish space Y Y such that E ∼B EG.

Proof. (3) =⇒ (1). Let E be a countable Borel equivalence on X satisfying (3). By Feldman-Moore X theorem E = EG for some countable G acting on X. Since F∞ surjects onto any countable group, there is an action of F∞ on X such that E = EF∞ . Then, by the assumption, there exists some Polish space Y and some free Borel action F∞ so that E ∼B YF∞ . It follows that E is treeable. (1) =⇒ (2). Assume that E induced by some Borel G-action and that E is treeable. Let (X, T) be a treeing for E. Fix a Borel linear order < on X, and put T+ = T ∩ <. There exists a Borel function ρ+ : T+ → G such that for all x, y, ρ+(x, y) · x = y. Next, define ρ : E → G by setting:

(i) ρ(x, x) = 1;

(ii) If x E y and x = x0 E ··· E xn = y is the unique path from x to y in (X, T), put ρ(x, y) = ∗ ∗ ∗ ρ (xn−1, xn)ρ (xn−2, xn−1) ··· ρ (x0, x1) where  ρ+(x, y) if (x, y) ∈ T+, ρ∗(u, v) = ρ+(x, y)−1 if (y, x) ∈ T+.

X (2) =⇒ (3). Assume that G is a countable group acting on X in a Borel way so that E = EG . Assuming (2) there exists a Borel cocycle ρ : E → G such that ρ(x, y) · x = y for all x E y. Now consider the action G on itself g · h = gh by left multiplication. For (x, g), (y, h) ∈ X × G define (x, g) ∼ (y, h) ⇐⇒ x E y and ρ(x, y)g = h.

Let Y = (X × G)/∼ with the quotient structure.

Claim. The relation ∼ is smooth, so Y is standard. (Exercise)

26 Define an action G y Y by −1 g · [x, h] = [y0, ρ(x, y0)hg ]

−1 for any y0 ∈ [x]E. Notice that if x E y0, this is well-defined because [y0, ρ(x, y0)hg ] = [x, hg−1]. Moreover, it is easily checked that the action is free.

Theorem 5.9 ([21]). If E is treeable, then E ≤B F(F2, 2). Hence F(F2, 2) is the universal treeable equiva- lence relation.

Question 5.10 (The finite index problem). Let E ⊆ F be coutable Borel equivalence relations such that E is treeable and every F-class contains only finitely many E-class. Is F treeable?

For some groups there are intrinsic restriction to induce a treeable equivalence relation. Those groups are called antitreeable.

X Theorem 5.11 (Adams-Spatzier [2]). If G is an infinite Kazhdan group acting on X so that E = EG and there is an E invariant probability measure on X, then E is not treeable.

Examples of Kazhdan groups include the special linear groups SLn(Z) and the projective spe- cial linear group PSLn(Z), which is the quotient of SLn(Z) modulo its center, for n ≥ 3. (E.g., see Lubotzky [29, Chapter 3].) n Let Rn be the equivalence relation induced from the action of GLn(Z) on T by

(aij) · (x¯1,..., x¯n) = (∑ a1jxj,..., ∑ anjxj). j j

Corollary 5.12. The equivalence relation Rn is not treeable for n ≥ 3.

6 Torsion-free abelian group of finite rank

Recall that an abelian group A is torsion-free if every nonzero element has infinite order, ie., na 6= 0 for all a ∈ A \{0} and n ∈ N. Let A be a countable abelian group, and P be the set of prime numbers.

Definition 6.1. The p-height of a is defined as

k ha(p) = sup{k ∈ N : there is x ∈ A such that p x = a} ∈ N ∪ {∞}.

27 Then the characteristic (or height sequence) χ(a) of a is defined to be

χ(a) = (ha(p) | p ∈ P).

Fact 6.2. (i) χ(a) = χ(−a);

( ) = ( ) = `i ` ≤ ∈ (ii) If χ a kp1 ,..., kpn ,... then a is divisible m ∏i∈I pi if and only if i kpi for i I.

(iii) χ(pna) = (ha(p1),..., ha(pn) + 1, . . . ). For this to make sense set ∞ + 1 = ∞.

(iv) Any sequence χ ∈ (N ∪ {∞})P can be realized as the height sequence of some elements in some 1 torsion-free abelian group. In fact, if A = h `n | `n ∈ N, `n ≤ χ(pn)i is a subgroup of Q such that pn χA(1) = (χ(p1), χ(p1),..., χ(pn),... ).

P Definition 6.3. Two h-sequences χ1, χ2 ∈ (N ∪ ∞) belong to the if and only if

(a) χ1(p) = χ2(p) for almost all primes.

(b) χ1(p) 6= χ2(p), then both χ1(p) and χ2(p) are finite.

We define the type t(a) of a as the class [χ(a)]≡.

Lemma 6.4. (a) If a and b are linearly dependent, then t(a) = t(b).

(b) If h : A → B is a group homomorphism, then t(a) ≤ t(h(a)).

Proof. To see (a) notice that t(a) = t(ma) for all m ∈ N. Thus if ma = nb for some m, n ∈ Z, then

t(a) = t(ma) = t(na) = t(b).

Recall that an abelian group has rank 1 if and only if every two nonzero element are linearly dependent. Hence we can define the type t(A) of A to be the ≡-equivalence class containing χ(a), where a is any non-zero element of A.

Theorem 6.5 (Baer [3]). Two torsion-free abelian groups of rank 1 are isomorphic if and only if they have the same type.

Proof. If a ∈ A and b ∈ B and t(a) = t(b) then there exist integers m, n such that χ(ma) = χ(nb). Then we can extend the partial function ma 7→ nb to an isomorphism A → B.

28 Definition 6.6. Let S(Q) ⊆ P(Q) = 2Q be the Polish space of torsion-free abelian group of rank 1. (S(Q) = {X ⊆ Q : X is a subgroup of Q} is closed in Q so S(Q) is a (compact) Polish space.)

The following is essentially a reformulation of Theorem 6.5.

∼ Proposition 6.7. =1 is Borel equivalent to E0.

Fuchs [13, Question 66] asked for a characterization of the invariant for the rank 2 torsion-free abelian groups. In order to address the question of Fuchs using techniques from the theory of countable Borel equivalence relation we introduce the relevant Borel spaces of rank 2 torsion-free abelian groups. n n n ∼ n Let S(Q ) = {A ⊆ Q : A ≤ Q } and =n be the isomorphism relation on S(Q ). As usual, denote by GLn(Q) the countable group of all invertible matrices n × n with entries in Q. It is clear 2 ∼ that GLn(Q) acts on S(Q ) by linear transformation and =n coincide with the orbit equivalence relation arising from this action, i.e., for A, B ∈ S(Qn)

∼ A =n B ⇐⇒ ∃M ∈ GLn(Q)(M · A = B).

Next theorem shows that such invariant are genuinely more complicated than the ones intro- duced by Baer in the case of torsion-free abelian groups of rank 1.

∼ ∼ ∼ Theorem 6.8 (Hjorth [17]). =2 is not hyperfinite. Hence, =1

∼ 2 We consider a subrelation of =2 restricted to a subspace of S(Q ) 2 2 2 2 2 2 2 Since SL2(Z ) ⊆ GL2(Q ) preserves Z , it acts on S(Q , Z ) = {A ∈ S(Q ) : Z ⊆ A}. Let S(Q2/Z2) = {A ⊆ Q2/Z2 : A ≤ Q2/Z2} be the standard Borel space of subgroups of 2 2 Q /Z . We consider the projective special linear group PSL2(Z) = SL2(Z)/{1, −1}. Since SL2(Z) is not amenable and {1, −1} is obviously amenable, PSL2(Z) is not amenable either. Instead of considering the action of GL2(Z) we consider the equivalence subrelation induced by the action of

PSL2(Z). (Notice that the SL2(Z) pushes down to a PSL2(Z)-action because 1 · A = −1 · A = A.) 2 2 2 2 2 2 Let π2 : Q → Q /Z be the canonical projection of Q onto Q /Z . The PSL2(Z)-action on 2 2 2 2 S(Q , Z ) projects down to a PSL2(Z)-action on S(Q /Z ). 2 2 For ease of exposition let PSL = PSL2(Z) and let S2 be S(Q /Z ). We are going to show that 2 S2 S(Q ) EPSL is not hyperfinite. Note that this will imply that EPSL is not hyperfinite as the point-wise 2 2 2 ∼ image map π2[−] : S(Q ) → S(Q /Z ) is a bijection. It will follow that =2 is not hyperfinite.

29 Let Γ = Q\2/Z2 = Hom(Q2/Z2, T) be the dual of Q2/Z2. We regard Γ with the point-wise

2 2 addition and the induced topology from TQ /Z . Γ is a compact Polish group. So we apply a classical theorem of Haar.

Theorem 6.9 (Haar). Every locally compact Hausdorff topological group admits a unique, up to a constant multiple, nontrivial left-invariant measure that is finite on compact sets.

Let µ be the Haar probability measure on Γ. Then let ν be the pushforward measure along the map ker: ψ 7→ ker(ψ). That is ν(B) = µ(ker−1[B]).

Lemma 6.10. ν is SL2(Z)-invariant.

Proof. Fix M ∈ SL2(Z).

ν(M · A) =µ(ker−1[M · A])

=µ({γ ∈ Γ : ker(γ) ∈ M · A})

=µ({γ ∈ Γ : {g : γ(g) = 0} ∈ M · A})

=µ({γ ∈ Γ : {M−1g : γ(g) = 0} ∈ A})

=µ({γ ∈ Γ : {g : γ(Mg) = 0} ∈ A}).

−1 Notice that the map ρM : Γ2 → Γ2, by ρM(γ)(x) = γ(M · x) is a homomorphic homeomorphism. So the last line of previous chain of equality is the same as

−1 µ({γ ∈ Γ : ker(ρM (γ)) ∈ A}) −1 =µ(ρM[ker [B]]) =

=µ(ker−1[B]]) =

=ν(B).

Lemma 6.11. There is an invariant measure 1 on which PSL2(Z) acts freely.

Proof. See [17, Lemma 5.3]

Proof of Theorem 6.12. Since PSL2(Z) is not amenable, ν is a PSL2(Z)-invariant probability mea- sure, and PSL2(Z) acts freely on a measure 1 invariant subset of S2, it follows by Theorem 4.26

S2 ∼ that EP SL is not hyperfinite. Therefore, =2 contains a non-hyperfinite equivalence relation and is not hyperfinite either.

30 Notice that every free action of PSL2(Z) induces a treeable equivalence relation. Hence our ∼ method does not say anything about the treeabiity of =2. In a similar way the following is a consequence of anti-treeability (see Theorem 5.11) and the fact that SLn(Z) is Kazhdan for all n ≥ 3.

∼ Theorem 6.12 (Hjorth [17]). =n is not treeable.

Definition 6.13. Let G be a locally compact second countable group. A subgroup H ⊆ G is said to be a lattice in G if it is discrete and there is an invariant Borel probability measure for the action of G on the cosets G/H by g · hH = ghH.

∼ Kechris sharpened Hjorth’s results by showing that =2 is not treeable. In his proof he presented

PSL2(Z[1/2]) as a lattice in a product group PSL2(R) × PSL2(Q2) and lifted a PSL2(Z[1/2]) in- variant probability measure to a measure invariant under the action of the product group. Note that both PSL2(R) and PSL2(Q2) contain an infinite amenable discrete group. E.g., consider    2k 0 | k ∈ Z 0 2−k

Theorem 6.14 (Kechris [24]). Suppose that H is a countable group which is a lattice in a product G1 ×

G2 of two locally compact second countable groups, each of which contains an infinite amenable discrete subgroup. If H is not amenable and acts freely on a standard Borel space X with an invariant probability X measure, then EH is not treeable.

7 Popa’s cocycle superrigidity

The cocycle superrigidity theorem of Popa is an instance of cocycle reduction result. We briefly discuss the idea behind it. First recall the notions of cocycle for Borel equivalence relations and group actions. The main difference from previous chapters is that we will be working in the measure theoretical setting. Until further notice let µ be a G-invariant probability measure on X.

Definition 7.1. Let E be a countable Borel equivalence relation on X. A map ρ : E → G is a cocycle if it satisfies the cocycle identity ρ(x, z) = ρ(y, z)ρ(x, y)

for µ-a.e. x, y ∈ X.

31 Definition 7.2. Two Borel cocycles ρ, σ : E → G are equivalent (in symbols ρ ∼ σ) if there is a Borel map f : X → G such that

x E y =⇒ σ(x, y) = f (y)ρ(x, y) f (x)−1 for µ-a.e. x, y ∈ X.

Definition 7.3. Let G be a countable group and a : G y X, a(g, x) = g · x be a Borel action. A Borel cocycle for a into H is a map α : G × X → H such that for all g0, g1 ∈ G,

α(g1g0, x) = α(g1, g0 · x)α(g0, x) for µ-a.e. x, y ∈ X

Example 7.4. Let a : G y X be a Borel action and h : G → H be a group homomorphism. Then we can simply define a cocycle by α(g, x) = h(g) for all x ∈ X.

Remark 7.5. If a : G y X is a Borel action. Any cocycle ρ : EG → H induces a cocycle α : G × X → H for a by setting α(g, x) = ρ(x, g · x).

If a : G → X is a free action, then any cocycle α : G × X → H gives rise to a cocycle ρ : EG → H by setting ρ(x, g · x) = α(g, x).

Definition 7.6. Two Borel cocycles α, β : G × X → H for a are equivalent (in symbols α ∼ β), if there is a Borel map f : X → H such that

β(g, x) = f (g · x)α(x, y) f (x)−1 for µ-a.e. x, y ∈ X.

We are now ready to state a special case of Popa’s superrigidity cocycle stating that under certain hypotheses on a group G acting on the space (X, µ), every Borel cocycle from G × X into an arbitrary countable group is equivalent to a group homomorphism. (A self-contained purely ergodic-theoretic presentation of Popa’s proof was given by Furman [14].)

Theorem 7.7 (Popa [38]). Let G ≤ G be countable groups such that G contains a Kazhdan infinite normal subgroup. If H is any countable group, then every Borel cocycle

α : G × (2)G → H for the shift action of G on (2)G is equivalent to a group homomorphism G → H.

In most applications we can let G = G. This will not be the case in the proof of Theorem 9.10. In order to explain how 7.7 can be used, consider the following situation. Let a : G y X and b : H y Y be Borel action on the standard Borel spaces X and Y respectively, and suppose that b is

32 free. Put E = Ea and F = Eb. Moreover, assume that ϕ : X → Y is a Borel homomorphism from E to F and α : G × X → H is the Borel cocycle of a associated to ϕ. That is, for every (g, x) ∈ G × X there is a unique α(g, x) ∈ H such that ϕ(g · x) = α(g, x) · ϕ(x). If α ∼ β via f : X → H, the perturbation ϕ0(x) = f (x)ϕ(x) of ϕ is also a homomorphism from E to F. Moreover, β is the cocycle associated to ϕ0. That is, ϕ0(g · x) = β(g, x) · ϕ0(x). Now if any cocycle α : G × X → Y for a is equivalent to a group homomorphism h : G → H and ϕ0 : X → Y is the homomorphism from E to F associated to h, then we have

ϕ0(g · x) = h(g) · ϕ0(x) (†)

If in a certain situation we can exclude that such ϕ0 as in (†) exists, then we can exclude the exis- tence of a Borel homomorphism from E to F at all. This will show that E is not Borel reducible to F in a very strong sense.

Applications of superrigidity

Let G be a countable acting on on a standard Borel space X in a Borel fashion. Suppose that µ is a G-invariant probability measure on X.

Definition 7.8. We say that the action of G on (X, µ) is ergodic (or G acts ergodically) if for every G-invariant Borel A ⊆ X either µ(A) = 1 or µ(A) = 0.

The following is a characterization of ergodic actions. For the proof and a more general state- ment see [4, Theorem 1.3].

Proposition 7.9. For a locally compact second countable topological group G the following are equivalent:

(i) G acts on (X, µ) ergodically;

(ii) If Y is a standard Borel space and f : X → Y is a G-invariant map (i.e., f (g · x) = f (x)), then there

is X0 ⊆ X with µ(X0) = 1 such that f  X0 is constant.

We introduce a stronger notion of ergodicity.

Definition 7.10. We say that the action of G on (X, µ) is strongly mixing if and only if for any two

Borel subsets A, B ⊆ X, if (gn)n∈N is a sequence of distinct elements of G, then

lim µ(gn(A) ∩ B) = µ(A)µ(B). n→∞

33 Notice that any strongly mixing action is necessarily ergodic. To see this, let A be a G-invariant

Borel subset of X. Then g(A) ∩ A = A for all g ∈ G. Hence for any sequence (gn)n∈N of distinct elements of G,

µ(A) = lim µ(A) = lim µ(gn(A) ∩ A) = µ(A)µ(A), n→∞ n→∞ therefore µ(A) is either 0 or 1.

Lemma 7.11. If G is a countable infinite group and µ is the product measure on (2)G, then the action of G on ((2)G, µ) is strongly mixing.

Proof. See [20, Proposition A6.1].

Note that if the action of G on (X, µ) is strongly mixing and H is an infinite subgroup of G, then the action of H is also strongly mixing, in particular H acts ergodically on (X, µ).

Definition 7.12. A Borel homomorphism from E to F is µ-trivial if there is a set X0 ⊆ X with

µ(X0) = 1 such that f maps X0 to a single F-class.

Definition 7.13. A virtual embedding between groups h : G → H is a homomorphism h : G → H with | ker h| < ∞.

Theorem 7.14. Let G = SL3(Z) × S, where S is any countable group. Suppose that H is any countable group and that Y is a free standard Borel H-space. If there exists a µ-nontrivial Borel homomorphism from Y F(G, 2) to EH, then there exists a virtual embedding h : G → H.

Proof. Let G = SL3(Z) × S and let H act freely on Y. Let E = F(2, G) and F = EH. Suppose that ϕ : (2)G → Y is a µ-nontrivial Borel homomorphism from E to F. Then we can define a Borel cocycle α : G × X → H by letting α(g, x) be the unique element of H such that

ϕ(g · x) = α(g, x) · ϕ(x).

By Theorem 7.7 we can suppose that α : G → H is a homomorphism, after adjusting ϕ if necessary. Now suppose for contradiction that K = ker α is infinite. Since the action of G on (2)G is strongly mixing, it is clear that K acts on (2)G ergodically. Moreover notice that the function ϕ : (2)G → Y is a K-invariant function. Thus, it follows that ϕ is constant µ-almost everywhere by Proposition 7.9. And this contradicts the assumption on ϕ.

Definition 7.15. A countable Borel equivalence relation E is free if there exists a countable group X G acting on X freely such that E = EG .

34 Theorem 7.16 (Thomas [43]). There are continuum many pair-wise incomparable free countable Borel equivalence relations

L∞ Proof. Let P be the set of primes and for each prime p ∈ P, let Ap = i=0 Cp be the direct sum of countably many copies of the cyclic group Cp of order p. For each subset C ⊆ P, let

M GC = SL3(Z) × Ap. p∈C

Claim 7.16.1. C ⊆ D if and only if F(GC, 2) ≤B F(GD, 2).

If C ⊆ D, then GC ≤ GDand hence F(GC, 2) ≤B F(GD, 2). Conversely, If F(GC, 2) ≤B F(GD, 2), then there exists a virtual embedding h : GC → GD. It is well-known that SL3(Z) contains a torsion-free subgroup of finite index. (E.g., see Wehrfritz [50, Corollary 4.8].) It follows that for L each p ∈ C, the cyclic group Cp embeds into q∈D Aq and this can be possible only if p ∈ D.

Definition 7.17. A countable Borel equivalence relation E is essentially free if there exists a free X countable Borel equivalence relation such that E ∼B EG .

Theorem 7.18 (Thomas [43]). If E is essentially free countable Borel equivalence relation, then there exists a countable group G such that F(G, 2) 6≤B E.

Y Proof. Let H be a countable group. Suppose that E ∼B EH for some free action of H on a standard Borel space Y. Let L be any finitely generated group that does not embed into H. (This existence of such group is guaranteed by a classical result of Neumann [35, Theorem 14] stating that there are uncountably many finitely generated group.) Let S = L ∗ Z and G = SL3(Z) × S. If F(G, 2) ≤ E then there is a virtual embedding π : G → H. Then π  S is a virtual embedding too. However L ∗ Z contains no normal notrivial subgroup, therefore | ker π  S| = 1. This implies that L embeds into H which is impossible.

Corollary 7.19. The class of essentially free countable Borel equivalence relations does not admit a universal element. In particular, E(F2, 2) is not essentially free.

We end this section stating other applications of Popa superrigidity theorem for countable Borel equivalence relations.

Theorem 7.20 (Thomas [43]). There exist uncountably many non-essentially free countable Borel equiv- alence relations that are incomparable up to Borel reducibility.

35 X Theorem 7.21 (Thomas [43]). Suppose that G is a quasi-finite group. Then EG is not universal.

Definition 7.22. Suppose that E and F are countable Borel equivalence relations on the standard Borel spaces X and Y, respectively. We say that E is weakly Borel reducible to F if there is a countable- to-one Borel homomorphism f : X → Y from E to F.

Theorem 7.23 (Thomas [43], after Adams [1] and Horth-Kechris [20]). There is a pair of countable Borel equivalence relation such that E ⊆ F and E and F are incomparable with respect to Borel reducibility.

Theorem 7.24 (Thomas [43]). If E is a weakly universal countable Borel equivalence relation, then E is not essentially free.

Y Proof. Otherwise, we can suppose that E = EH for some free Borel action H y Y. But then there Y exists a countable G such that every homomorphism from F(G, 2) to EH maps a measure one set Y Y into a single EH-class. Therefore, EG is not weakly Borel reducible to EH.

8 (Anti)classification of torsion-free abelian groups of finite rank

In this section we discuss the proof of the following result:

Theorem 8.1 (Thomas [42]). For all n ≥ 3,

∼ ∼ =n−1

Theorem 8.1 is a consequence of the following result.

Theorem 8.2 (Thomas [42]). Let n ≥ 3. Suppose that SLn(Z) acts on a standard Borel space X and µ is m an ergodic SLn(Z)-invariant probability measure. If 1 ≤ m < n and f : X → S(Q ), x 7→ Ax is a Borel X ∼ homomorphism from E to =m, then there exists a SLn(Z)-invariant Borel M ⊆ X such that f maps SLn(Z) ∼ M into a single =m-class.

Exercise 8.3. Suppose that a countable group G is acting on a standard Borel space X in a Borel fash- ion and µ is an ergodic G-invariant probability measure on X. If Y is a a standard Borel space, H is a X Y countable group acting on Y in a Borel fashion, f : X → Y is a Borel homomorphism from EG to EH, and

{StabH( f (x)) : x ∈ X} is countable. Then there exists a fixed subgroup L ≤ G, a G-invariant X1 ⊆ X

0 X1 0 with µ(X1) = 1, and a Borel homomorphism f : X1 → Y from EG to EH such that StabH( f (x)) = L, for all x ∈ X1.

36 In the scenario described by Exercise 8.3 the equivalence relation EH  f (X1) will be induced by a free action of the quotient group H = NH(L)/L on the Borel set Y = {y ∈ Y : Stab(y) = L}.

This will allow us to define a cocycle α : G × X1 → NH(L)/L where NH(L) = {h ∈ H : hL = Lh} is the normalizer of L in H. So we might be able to apply a suitable cocycle reduction theorem. Unfortunately, the situation of Theorem 8.2 is way different than the idyllic landscape described ∼ above. The isomorphism relation =m is induced by the action of GLm(Q) and unfortunately there are uncountably possibilities for Stab(Ax) = Aut(Ax). This explains why the proof of Theorem 8.2 is quite involved. To come across this difficulty we shall consider a coarser equivalence relation on S(Qn).

Quasi-equality and quasi-isomorphism

Definition 8.4. Suppose that A, B ∈ S(Qm). We say that A and B are quasi-equal (in symbols

A ≈m B) if and only if A ∩ B has finite index in both A and B.

Definition 8.5. Suppose that A, B ∈ S(Qm). We say that A and B are quasi-isomorphic (in symbols

A ∼m B) if and only if there is ϕ ∈ GLn(Q) such that ϕ · A ≈m B.

Proposition 8.6. For all n > 0, ≈n is a countable Borel equivalence relation. Then, so is ∼n.

n n Proof. Given A ∈ S(Q ) there are only countably many B ∈ S(Q ) such that A ≈n B. To see this, notice that if A ≈n B then there are finite integers r, s > 0 such that rA ≤ B and sB ≤ A. Then B satisfies necessarily 1 rA ≤ B ≤ A s 1 and [ s A : rA] = [A : rsA] < ∞. There are only countably many possibilities for B.

In the proof of Theorem 8.2 we shall consider the action of GLm(Q) on the ≈m-classes. First, we need to give a characterization of the stabilizer for this action.

Definition 8.7. A linear transformation ϕ ∈ Matm(Q) is a quasi-endomorphism of A if and only if

ϕ(A) ≺m A. Equivalently, if and only if there exists k > 0 such that kϕ ∈ End(A).

Fact 8.8. Let A, B ∈ S(Qm).

• End(A) is a Q-subalgebra of Matm(Q),

• If A ≈m B, then QE(A) = QE(B).

37 Definition 8.9. A linear transformation ϕ ∈ Matm(Q) is a quasi-automorphism of A if and only if ϕ is a unit of the Q-algebra QE(A).

Fact 8.10. Let A ∈ S(Qm) and ϕ ∈ End(A)

• QAut(A) is a group,

• ϕ ∈ QAut(A) if and only if ϕ is a monomorphism.

Lemma 8.11 (Thomas [42, Lemma 3.6]). If A ∈ S(Qm), then QAut(A) is the (set-wise) stabilizer of

[A] in GLn(Q).

Proof of Theorem 8.2

Clearly x E y implies Ax ∼m Ay, hence f is a Borel homomorphism from E to ∼m. By Lemma 8.11 it follows that |{QE(Ax) : x ∈ X}| ≤ ω because there are only countably many possibilities for the Q-subalgebras of Matm(Q).

Therefore there is a SLn(Z)-invariant Borel subset X1 ⊆ X with µ(X1) = 1 and after possibily adjusting f there is a fixed Q-subalgebra S of Matm(Q) such that QE(Ax) = S for all x ∈ X1. For ∗ the sake of readability we put X = X1. In particular, for all X = X1, QAut(Ax) = S the group of units of S.

Now, for all x, y ∈ X such that x E y, Ax ≈ Ay. So there is ϕ ∈ GLn(Q) such that ϕ(Ax) ≈m Ay. ∈ ( ) Note that ϕ NGLm(Q) S as

−1 −1 ϕSφ = ϕQE(Ax)φ = QE(ϕ(Ax)) = QE(ϕ(Ay)) = S.

= ( ) ∗ = { ∗ ∈ } = ∗ Let H NGLn(Q) S /S ϕS : ϕ NGLn(Q) . For readability let ϕ¯ ϕS . Notice that H acts on f (X1)/≈m freely by ϕ¯ · [A] = [ϕ(A)]. (To see this, observe that ϕ([A]) = ϕ¯ · [A] = [A] implies that ϕ ∈ QAut(A) by Lemma 8.11, thus ϕ ∈ S∗.)

Then we can define a Borel cocycle α : SLn(Z) × X → H by letting α(g, x) be the unique ϕ¯ ∈ H such that ϕ¯ · [Ax] = [Ag·x]. Now we are ready to apply the following reduction theorem:

Theorem 8.12 (Thomas [42]). Let n ≥ 3. Suppose that SLn(Z) acts on a standard Borel space X and µ 2 is an ergodic SLn(Z)-invariant probability measure. If G is an algebraic Q-group with dim G < n − 1 and H ≤ G(Q). Then every Borel cocycle α : SLn(Z) × X → H is equivalent to a Borel cocycle β such that β(SLn(Q) × X) is contined in a finite subgroup of H.

38 The fact that such algebraic Q-group exists is nontrivial (see [42, Lemma 3.7] for details).

So let β : SLn(Z) × X → H a cocycle such that β(SLn(Z) × X) is contained in a finite subgroup

K < H. Also let b : X → H be a Borel map such that for all g ∈ SLn(Z)

α(g, x) = b(g · x)β(g, x)b(x)−1 for µ-a.e. x ∈ X.

Theorem 8.13 (Hjorth-Kecris [19, Theorem 10.5]). Suppose that a countable infinite Kazhdan group G acts on a standard Borel space X and µ is an ergodic G-invariant probability measure. If F is any treeable X equivalence relation on Y and f is a Borel homomorphism from EG to F then there exists a Borel X0 ⊆ X such that µ(X0) = 1 and f maps X0 into a unique F-class.

k For all k > 2, and (A1,..., Ak), (B1,..., Bk) ∈ (S(Q)) define

(A1,..., Ak) F (B1,..., Bk) ⇐⇒ {[A1],..., [Ak]} = {[B1],..., [Bk]}.

Theorem 8.14 (Thomas [42, Theorem 3.8]). For all n > 0, the countable Borel equivalence relation ≈n is hyperfinite.

Corollary 8.15. For all n > 0,Fn is hyperfinite.

Lemma 8.16. There exist a SLn(Z)-invariant X1 ⊆ X with µ(X1) = 1, a fixed k > 0, and a Borel homomorphism f˜: X → (S(Qm))k, from EZ to F such that for every x, the groups in f˜(x) are all 1 SLn(Z) k quasi-isomorphic to Ax.

Lemma 8.16 together with theorem 8.13 gives that f˜ maps X1 into a unique Fk-class. There- m fore the is a ∼m-class C ⊆ S(Q ) such that f (X1) ⊆ C. One more application of ergodicity gives

Theorem 8.2. We already observed that ∼m is a countable Borel equivalence relation. (See Propo- sition 8.6). Hence there exists and some Z ⊆ X1 with µ(Z) > 0 and some group A ∈ C such that f (x) = A for each x ∈ Z. Hence let M = SLn(Z) · Z. Since M is SLn(Z)-invariant, µ(M) = 1 by ∼ ergodicity. Moreover f (M) is contained in the =m-class of A. In the rest of this section we discuss the classification problem for finitely generated torsion- free abelian groups.

Definition 8.17. If Ei, i < n, where n ≤ ∞, are Borel equivalence relations, with Ei living on Xi, L F S then we let i

M (x, j) Ei (y, k) ⇐⇒ j = k and x Ej y. i

39 L ∼ Theorem 8.18 (Thomas[43]). n =n is not universal.

Proof. Let S be a suitable countable group, and let G = SL3(Z) × S. Consider the free action of G on ((2)G, µ) where µ is the usual product probability measure. Suppose that f : (2)G → F n L ∼ n S(Q ), x 7→ Ax is a Borel reduction from F(G, 2) to n =n. By ergodicity there exists a G- G n invariant subset X1 ⊆ (2) such that f (X1) ⊆ S(Q ) for a fixed n < ω. For ease of exposition G n we assume that f : (2) → S(Q ). As in the proof of Theorem 8.2, the action of GLn(Q) is not free. Again, we get around this difficulty by considering the coarser equivalence relation of quasi- isomorphism. ∈ ( )G ( ) = ([ ]) For each x 2 , there are countably many possible values for QAut Ax StabGLn(Q) Ax . G Then, by ergodicity there is a fixed subgroup L ≤ GLn(Q) and some Borel X1 ⊆ (2) , with

µ(X1) = 1, such that QAut(Ax) = L for all x ∈ X1. ∈ ( )G ∈ ( ) = { ∈ = } If x, y 2 such that x EG y, then there is ϕ NGLn(Q) L ψ GL : ψL Lψ such that

ϕ · [Ax] = [Ay]. So let H = N/L. We can define a Borel cocycle α : G × (2)G → H by setting α(g, x) → H the unique ϕ ∈ H such that ϕ · [Ax] = [Ag·x]. At this point we specify the nature of S. Let S be a simple nonamenable countable group such that S does not embed into any of the possible values for H. By Theorem 7.7, α is equivalent to a Borel homomorphism. We can assume that α : G → H is a group homomorphism possibly deleting a null set. By the assumption on S, we have S ≤ ker α. It follows that for every g ∈ S

G [Ag·x] = [Ax] µ-a.e. x ∈ (2) .

Therefore, up to a measure 0 set f is a Borel homomorphism from ES to ≈n. Now we recall the following result.

Theorem 8.19 (Losert-Rindler [28], Jones-Schmidt [22]). If Γ ≤ ∆ is a nonamenable group then the ∆ shift action Γ y 2 is E0-ergodic.

Proof. See [20, Theorem A4.1].

By Theorem 8.14 the quasi-equality relation ≈n is hyperfinite. It follows that there is a subset G of (2) of measure 1 that is mapped by f into a single ≈n-class. This is a contradiction.

Notice that the following is an open question and an affirmative answer will imply an alterna- tive prove of Theorem 8.18.

40 ∼ Question 8.20. Is =n essentially free, for n ≥ 2?

9 The algebraic structure of bireducibility degrees

In this section we are concerned with the structural properties of countable Borel equivalence rela- tions. We show that the class C of countable Borel equivalence relations modulo Borel reducibility carries a rich additional algebraic structure.

Definition 9.1 (Tarski [41]). A cardinal algebra is a triple hA, +, ∑i such that:

(I) hA, +i is an abelian monoid;

(II) Σ : AN → A is an infinitary operation such that:

(i) ∑n an = a0 + ∑n an+1;

(ii) ∑n(an + bn) = ∑n(an) + ∑n(bn)

N (iii) If a + b = ∑n cn, then there are (an), (bn) ∈ A with a = ∑n an and b = ∑n bn such that

cn = an + bn.

N (iv) If (an), (bn) ∈ A with an = bn + an+1, then there is c such that for each n

an = c + ∑ bn+i. i

Cardinal algebras were introduced by Tarski to axiomatize addition of cardinals without in absence of the choice axiom.

Remark 9.2. When hA, +, ∑i is a cardinal algebra, we can define a partial order on A by declaring a ≤ b if and only if ∃c(a + c = b).

Given Borel equivalence relations E, F on standard Borel spaces X and Y respectively, we say i that E invarianlty embeds in to F (written E vB F) if there is a Borel F-invariant A ⊆ Y so that ∼ E =B F  A.

Definition 9.3 (Kechris-McDonald [25]). A class of Borel equivalence relation is Tarskian if and only if

(1) ∅ ∈ E.

41 i (2) If F ∈ E and E vB F, then E ∈ E.

L (3) If F0, F1,... ∈ E, then n Fn ∈ E.

L ∼ (4) If E, F0, F1,... ∈ E and E ∼B n Fn, then there are En ∈ E with Fn ∼B En such that E =B L n En.

(5) E, F ∈ E live on X and Y, respectively. Then f : X → Y, E ≤B F. Then the saturation

A = [ f (X)]F is a Borel subset of Y and E ∼B F  A.

If E is a class of Borel equivalence relations, let [E] = {[E] : E ∈ E}, where [E] = {F ∈ E : E ∼B L F}. Then if E, F ∈ E let [E] + [F] = [E ⊕ F]. Moreover, if Fn ∈ E, let ∑n[Fn] = [ n Fn].

Exercise 9.4. If E is a Tarskian class of Borel equivalence relation then h[E], +, ∑i is a cardinal algebra.

Moreover, for E, F ∈ E we have that E ≤B ⇐⇒ [E] ≤ [F].

Corollary 9.5. h[C], +, ∑i is a cardinal algebra.

Let B be the class of all Borel equivalence relations.

Question 9.6. Is h[B], +, ∑i a cardinal algebra?

Remark 9.7. For E, F ∈ B it does NOT hold in general that E ≤B F ⇐⇒ [E] ≤ [F].

Theorem 9.8 (Hjorth [18]). There is a Borel equivalence relation E that Borel reduces to some CBER F but is not Borel equivalent to any CBER.

It follows that [E] 6≤ [F]. For suppose that [E] ≤ [F] and let H be such that E ⊕ H ∼B F. Let f : Y → X t Z be a Bore reduction from F to E ⊕ H. Then A = f −1(X) would be an F-invariant subset of Y, and E ∼B F  A, which contradicts Theorem 9.8. Corollary 9.5 has several consequences on the stracture of C.

(A) Existence of suprema. Any increasing sequence F0, F1, . . . admits a least upper bound in ≤B

(B) Interpolation. (Interpolation) If S, T are countable sets of countable Borel equivalence re- lations and every E ∈ S is Borel reducible to each F ∈ T , then there is a countable Borel

equivalence relation F0 such that for all E ∈ S, F ∈ T

E ≤B F0 ≤B F.

42 (C) Cancellation. If n > 0 and E, F are countable Borel equivalence relations, then

nE ≤B nF =⇒ E ≤B F,

thus

nE ∼B nF =⇒ E ∼B F.

Definition 9.9. If E and F are Borel equivalence relations on X and Y respectively, then the product E × F is an equivalence relation on X × Y. Precisely,

(x, y) E × F (x0, y0) ⇐⇒ x E x0 and y F y0.

If E is a Borel equivalence relations on X, we denote by En = X × · · · × X. Notice that h[C], ·i | {z } n times is an abelian monoid with [E] · [F] = [E × F] and identity element [=1], the equality relation on a space of one element.

2 2 Theorem 9.10 (Kechris-Macdonald [25]). There are CBERs E

Lemma 9.11. Suppose that R, S are Borel equivalence relations on (X, µ) and (Y, ν) respectively such that

(i) R and S have infinitely many equivalence classes;

2 2 (ii) R ∼B R and S ∼B S

(iii) R and S are µ-ergodic and ν-ergodic respectively, and for all R-invariant Borel A ⊆ X, with µ(A) =

1, we have R  A 6≤B S. (Similarly, for all S-invariant Borel C ⊆ Y, with µ(A) = 1,S  C 6≤B R.)

2 2 Then, if E = R ⊕ S and F = R × S,E

Proof. First, we prove that E ≤B F. It follows by Exercise 3.2 that for fixed x0 ∈ X and y0 ∈ Y,

E = R ⊕ S ≤B R  (X r [x0]R) ⊕ S  (Y r [y0]S). (1)

Let Z = (X r [x0]R) t (Y r [y0]S). Then the function f : Z → X × Y, x 7→ (x, y0), y 7→ (x0, y) is a Borel reduction from the latter equivalence relation in (1) to R × S = F. 2 2 2 2 2 2 Next, we prove that E ∼B F . Since F = (R × S) ∼B R × S ∼B R × S, it suffices to show 2 2 that E = (R ⊕ S) ∼B R × S.

Last, we show that E

Now notice that both X0 and Y0 are R × S-invariant.

43 Claim.R × S is µ × ν-ergodic.

It follows that either µ × ν(X0) = 1 or µ × ν(X0) = 0 (which implies µ × ν(Y0) = 1). In the first case, there is x ∈ X such that ν((X0)x) = 1. Also note that the vertical section (X0)x is S-invariant. Then the function

(X0)x → X t Y

y 7→ f (x, y)(∈ X) witnesses that S  (X0)x ≤B R. This is contradictory with (iii). If µ × ν(Y0) = 1, we argue in a similar way.

Proof of Theorem 9.10. It suffices to find two countable Borel equivalence relations R are S that sat- isfy conditions (i)–(iii) from Lemma 9.11. The proof of this fact uses the following result.

Theorem 9.12 (essentially Ol’shanskiˇ [36]). There is a countable Kazhdan, torsion free, simple group Γ, and a countably infinite Kazhdan, torsion, simple group ∆.

Given Γ and ∆ as above, define Γ∗ = Γ ⊕ Γ ⊕ · · · and ∆∗ = Γ ⊕ Γ ⊕ · · ·. Notice that any homomorphism Γ → ∆∗ or ∆ → Γ∗ is trivial. Now put R = F(Γ∗, [0, 1]) and S = F(∆∗, [0, 1]) and ∗ ∗ let µ and ν the usual product measure on ([0, 1])Γ and ([0, 1])∆ , respectively. It is clear that (i) holds. To see (ii) it is enough to show that R2. Notice that R2 coincide with the equivalence relation ∗ ∗ on ([0, 1])Γ × ([0, 1])Γ induced by the free action of Γ∗ × Γ∗ defined by (γ, δ) · (x, y) = (γ · x, δ · y). Since all uncountable standard Borel space are isomorphic, there is a free action a : Γ∗ × Γ∗ y [0, 1] ∼ 2 inducing a countable Borel equivalence relation Ea = R . It is a general fact that if a countable group G y X freely, then the map X → XG, x 7→ (g 7→ −1 X g · z) witness that EG ≤B F(G, [0, 1]). So we have

2 ∼ ∗ ∗ ∼ ∗ R =B Ea ≤B F(Γ × Γ , [0, 1]) =B F(Γ , [0, 1]) = R.

2 2 Hence R ∼B R and similarly S ∼B S . For (iii) notice that R and S are µ-ergodic and ν-ergodic respectively. It remains to show that if

A ⊆ is R-invariant and µ(A) =, then R  A 6≤B S.

Suppose that f : A → Y witnesses R  A ≤B S. Then for every g ∈ Γ, x ∈ A there is unique α(g, x) ∈ ∆∗ such that f (g · a) = α(g, x) f (x). Thus the function α : G × X → ∆∗ is a Borel cocycle.

44 By Popa’s superrigidity theorem (see Theorem 7.7) there is a (necessarily trivial) group homomor- phism h : Γ → ∆∗ and a Borel reduction b : X → ∆∗ such that

α(g, x) = b(g · x)(g · x)h(x)b(x)−1 µ − a.e. x ∈ A.

Now consider the function f 0(x) = b(x)−1 · f (x). 0 Notice that, up to a null set, f is Borel reduction from R  A to S. Moreover, it is not hard to see ∗ that f 0 is G-invariant (i.e., f 0(g · x) = f 0(x)). By ergodicity of the shift action of Γ y ([0, 1]Γ , µ), 0 it follows that f is constant on some X0 ⊆ A with µ(X0) = 1. Clearly the latter condition is incompatible with f 0 being a Borel reduction, because Borel reductions are one-to-one.

10 Martin’s conjecture

In this section we discuss a crucial interplay between recursion theorey and descriptive set theory.

N Definition 10.1. For r, s ∈ 2 we say that r is Turing reducible (written r ≤T s) if and only if there exists an oracle Turing machine that computes r when its oracle tape contains s.

We say that r and s are Turing equivalent (written r ≡T s) iff r ≤T s and s ≤T r.

It is clear from the definition that ≡T is a countable Borel equivalence relation. In fact, every N point s ∈ 2 has acountably many ≤T-predecessors.

Definition 10.2. For each r ∈ 2N, the corresponding cone is the Borel subset C ⊆ 2N defined by

N C = {s ∈ 2 | r ≤T s}.

N Theorem 10.3 (Martin). If A ⊆ 2 is a ≡T-invariant Borel subset, then either A contains a cone or 2N r A contains a cone.

Proof. The proof uses Martin’s result that every Borel game is determined. Let A ⊆ 2N be a Borel

≡T-invariant set. Consider the Gale-Stewart game G(A) with payoff with payoff set A. That is, the game described by the following plays.

I s0 ∈ {0, 1} s2 ∈ {0, 1}··· s2n ∈ {0, 1}

II s1 ∈ {0, 1}··· s2n+1 ∈ {0, 1} where Player I wins if and only if s = (sn)n∈N ∈ A. By Borel determinacy theorem, one of the two players has a winner strategy.

45

It follows that C ⊆ A. To see this, let t ∈ C so we have σ ≤T t. Now consider the following plays for G(A)

_ • Player I plays some s0 ∈ {0, 1}, and then follows the winning strategy σ (i.e., s2n = σ(s0 _ ··· s2n−1));

• Player II plays s2n+1 = tn.

Since σ is a winning strategy for Player I, it follows that s ∈ A. Moreover, it is clear that t ≤T s.

On the other hand, we have σ ≤T t and s ≤ σ ⊕ t, which implies that s ≤T t. So s ≡T t. Since A is

≡T-invariant, we have t ∈ A.

N N Corollary 10.4. If f : 2 → 2 is a Borel function such that x ≡T y implies f (x) = f (y), then there is a N cone C ⊆ 2 such that f  C is constant.

N Proof. It follows by Theorem 10.3 that for each n ∈ N, either {x ∈ 2 : f (x)n = 0} or {x ∈ N 2 : f (x)n = 1} contains a cone. So, for each n ∈ N, there is en ∈ {0, 1} such that Xn = {x ∈ N T 2 : f (x)n = en} contains a cone Cn. Let D = n Cn. Clearly f  D is constant. Moreover, we T N N notice that n Cn contains a cone. To see this let rn ∈ 2 be such that Cn = {s ∈ 2 : rn ≤T s}. L T L Then let n rn be the join of the rn’s and let D = n Cn. Clearly, rn ≤T n rn. It follows that N L C = {s ∈ 2 : n rn ≤T s} ⊆ D.

N N 0 0 We say that A ⊆ 2 is ≤T-cofinal if for every s ∈ 2 there is s ∈ A such that s ≤T s .

N Exercise 10.5. Let A ⊆ 2 be a ≡T-invariant Borel subset. If A is ≤T-cofinal then A contains a cone.

N N Definition 10.6. A Turing invariant function is a Borel homomrphism f : 2 → 2 from ≡T to itself.

The follwoing open problem is the Borel version of the first part of a conjecture of Martin [31].

Conjecture 10.7 (MC). If f : 2N → 2N is a Turing invariant Borel function, then exactly one of the following holds:

N (I) There is a cone C ⊆ 2 , f maps C into a single ≡T-class;

N (II) There is a cone C ⊆ 2 such that x ≤T f (x) for all x ∈ C.

46 Remark 10.8. Without the requirement that f be Borel the statement above is false in ZFC. In fact, the axiom of choice implies the existence of several pathological Turing invariant functions. The original version of Martin’s conjecture2 is formulated in ZF + DC + AD. (See also the excellent introduction of [32].)

The only known progress towards MC is the following result by Slaman and Steel.

Theorem 10.9 (Slaman-Steel [40]). Suppose that f : 2N → 2N is a Turing invariant Borel function. If N there is a cone C ⊆ 2 such that f (x)

into a single ≡T-class.

(MC0) If f : 2N → 2N is a Turing invariant Borel function, then exactly one of the following holds:

N (I) For all x ∈ 2 , there is x ≤T y such that f (y)

N (II) For all x ∈ 2 , there is x ≤T y such that y ≤T f (y).

The following observation will be useful later in this section.

Exercise 10.10. (MC) is equivalent to (MC0)

The Borel complexity of ≡T

In this subsection we investigate some consequences of MC. First it is natural to ask what is the

Borel complexity of ≡T.

2 Theorem 10.11 (MC). (≡T) 6≤B ≡T. Thus, ≡T is not universal.

N N N 2 N Proof. Suppose that f : 2 × 2 → 2 is a Borel reduction from (≡T) to ≡T. Let r, s ∈ 2 be such N N N N that r 6≡T t. Define f0 : 2 → 2 , s 7→ f (r, s) and f1 : 2 → 2 , s 7→ f (t, s). Since r and t are not N N ≡T-equivalent, it is clear that A0 = [ f0(s )] and A1 = [ f1(2 )] are disjoint ≡T-invariant Borel set.

Also, for each i ∈ {0, 1}, it cannot be the case that fi is constant on a cone. Hence, there is a N cone Ci ⊆ 2 such that x ≤T fi(x) for all x ∈ Ci. This implies that each Ai is ≤T-cofinal, then each

Ai contains a cone by Exercise 10.5. This contradicts the fact that A0 and A1 are disjoint.

Although the exact Borel complexity of ≡T is unknown. One can prove that ≡T is universal with respect to weak Borel reducibility. (cf. Definition 7.22.)

2Martin’s conjecture is one of the few unsolved Victoria Delfino’s problem. More on the character of Victoria Delfino and the list of problems entitled to her can be found in [5].

47 Definition 10.12. A countable Borel equivalence relation F is weakly universal if every countable w Borel equivalence relation E is weakly Borel reducible to F (in symbols, E ≤B F).

The free group F2 can be identified with a suitable group of recursive permutations of N. So the orbit equivalence relation E(F2, 2) is contained in ≡T. This implies that ≡T is weakly universal.

N N Theorem 10.13 (MC). If f : 2 → 2 is a Borel ≡T-invariant function, then either

N (i) There is a cone C ⊆ 2 such that f maps C into a single ≡T-class;

N (ii) There is a cone C ⊆ 2 such that f  C is a weak Borel reduction from ≡T  C to ≡T. Moreover, for N every cone D ⊆ 2 the saturation [ f (D)]≡T contains a cone.

N Proof. If (i) fails, then (MC) implies the existence of C ⊆ 2 such that x ≤T f (x) for all x ∈ C. N Since every x ∈ 2 has countably many predecessors f  C is necessarily countable-to-one. So, N f  C is a weak Borel reduction from ≡T  C to ≡T. Moreover, let D ⊆ 2 be any cone and let

D0 = D ∩ C. Since f  C is countable-to-one, it follows that f (D0) is Borel. (See Exercise 1.16.)

Therefore, the saturation [ f (D0)]≡T is a Borel ≡T-invariant set. Moreover, [ f (D0)] is ≤T-cofinal. It follows by Exercise 10.3 that [ f (D0)]≡T contains a cone.

N Corollary 10.14 (MC). If A ⊆ 2 is a ≡T-invariant Borel subset, then ≡T  A is weakly universal if and only if A contains a cone.

N Proof. If ≡T  A is weakly universal, then there is a weak Borel reduction h : 2 → A from ≡T to N ≡T  A. Then condition (i) of Theorem 10.13 must hold and [h(2 )]≡T contains a cone. Conversely, suppose that the cone C = {s ∈ 2N : r ≤ s} is contained in A. Given two reals x, y ∈ 2N, the join x ⊕ y is the real x ⊕ y defined (x ⊕ y)(2n) = x(n) and (x ⊕ y)(2n + 1) = x(n) for all n ∈ N. Then, the map s 7→ s ⊕ r is an injective weakly Borel reduction from ≡T to ≡T A.

"Ergodicity“ for Turing equivalence

Definition 10.15. Let E be a countable Borel equivalence relation on a countable Borel space X. N We say that ≡T is E-m-ergodic if for every Borel homomorphism f : 2 → X from ≡T to E, there is a cone C ⊆ 2N such that f maps C into a single E-class.

N Notice that Corollary 10.4 can be reformulated by saying that ≡T is id(2 )-m-ergodic. Cur- rently, we do not know any example of nonsmooth countable Borel equivalence relation E such that ≡T is E-m-ergodic. In particular the following question is open.

48 Question 10.16. Is ≡T E0-m-ergodic?

Theorem 10.17 (MC). If E is a countable Borel equivalence relation then either one of the following is true:

(a) E is weakly universal; or

(b) ≡T is E-m-ergodic.

Proof. If E is weakly universal, then there is a countable-to-one Borel homomorphism from ≡T to E. Hence (b) is false. N Next, suppose that (b) fails. Then, there is a Borel homomorphism ϕ : 2 → X from ≡T to E N such that for every cone C ⊆ 2 , ϕ(C) is not contained into a single E-class. Since ≡T is weakly N universal, there is a weak Borel reduction ψ : X → 2 from E to ≡T. Then θ = ψ ◦ ϕ is a Borel

≡T-invariant map from ≡T to ≡T. Thus, Martin’s conjecture implies that θ satisfies either (i) or (ii) of Theorem 10.13. N We claim that (i) cannot hold. To see this, suppose that there is a cone C0 ⊆ 2 such that θ −1 maps C into a single class [x]≡T . Then, Y = ψ ([x]≡T ) is countable because ψ is countable-to-one, −1 and ϕ maps C0 into a Y. Notice that there is y ∈ Y such that Z = ϕ (y) is a ≤T-cofinal Borel N subset of 2 . (Why? Exercise.) Then, Exercise 10.5 implies that the saturation [Z]≡T contains a cone D. It follows that D is mapped by ϕ into a single E-class, which contradicts the assumption that ϕ witnesses the failure of (b). N So, Theorem 10.13 (ii) implies that there is a cone C ⊆ 2 such that θ  C is a weak Borel reduction from ≡T C to ≡T. In particular, θ  C is a countable-to-one map, and ϕ  C is necessarily w countable-to one. We conclude that (≡T  C) ≤B E, which implies that E is universal as desired.

Exercise 10.18 (MC). Let E, F be countable Borel equivalence relations on X, Y, respectively. Suppose that E is weakly universal and that F is not weakly universal. If ϕ : X → Y is a Borel homomorphism from E to

F, then there exists a Borel X0 ⊆ X such that:

(i) E  X0 is weakly universal

(ii) ϕ maps X0 into a single F-class.

Definition 10.19. Let G be a countable group. We say that G is weakly action universal if there is a X Borel G-action on some standard Borel space X such that EG is weakly universal.

49 Definition 10.20. Let ≈G be the conjugacy equivalence relation on S(G) = {H ⊆ G : H ≤ G}, the standard Borel space of subgroups of G. That is, given H, K ≤ G, we say that

−1 H ≈G K ⇐⇒ ∃g ∈ G (K = gHg ).

Theorem 10.21 (Thomas [45]). (MC) If G is a countable group then the following are equivalent:

(i) G is weakly action universal;

(ii) ≈G is weakly universal.

X Proof. Clearly, (ii) implies (i). Suppose that G → X, and EG is weakly universal and ≈G is not weakly universal.

Consider the stabilizer Borel map σ : X → S(G), σ(x) = Gx = {g ∈ G : g · x = x}. Notice that X −1 σ is a Borel homomorphism from EG to ≈G as, whenever g · x = y, then Gy = g · Gx g . Now, X N X since EG is weakly universal, there exists a weak Borel reduction ϕ : 2 → X from ≡T to EG . So

θ = σ ◦ ϕ is a Borel homomorphism from ≡T to ≈G. It follows from Theorem 10.17 that there is a N cone C ⊆ 2 such that θ(C) is contained into a single ≈G-class. Up to redefining φ, we can assume that there is a fixed K ≤ G such that Gϕ(x) = K for each x ∈ C. Let X0 = {x ∈ X : Gx0 = K}. w X X Therefore, we have ≡T  C ≤B EG  X0, and this implies that EG  X0 is weakly Borel universal because so is ≡T  C. (Cf. Corollary 10.14.)

Notice that for all x, y ∈ X0, whenever g · x = y for some g ∈ G, then g ∈ NG(K) = {g ∈ X G : Kg = gK}. Then EG  X0 is the equivalence relation induced by the obvious free action of X NG(K)/K on X0. It follows by Theorem 7.24 that EG  X0 is not weakly universal. A contradiction.

Uncountably many weakly universal CBERs

Theorem 10.22 (Thomas [44]). (MC) There exist uncountable many weakly universal countable Borel equivalence relations up to Borel bi-reducibility.

Definition 10.23. Let E, F be equivalence relations on the standard Borel spaces X, Y, respectively. Suppose that µ is a E-invariant probability measure on X. Then E is F-ergodic if and only if for every Borel homomorphism f : X → Y from E to F, there is a Borel Z ⊆ X with µ(Z) = 1 such that f maps Z into a single F-class.

Notice that ergodicity defined as in Definition 10.23 transfers to the measure theoretical con- text.

50 Exercise 10.24. If E is F-ergodic then for every µ-measurable homomorphism from E to F, there is a Borel Z ⊆ X with µ(Z) = 1 such that h maps Z into a single F-class.

N Fact 10.25. There is an uncountable family G = {Gα : α ∈ 2 } of finitely generated groups such that:

(a)SL 3(Z) / Gα;

(b) Gα has no nontrivial finite normal subgroups;

(c) if α 6= β then Gα does not embed into Gβ.

N Moreover, we can assume that each of the Gα is a group on N and that {Gα : α ∈ 2 } forms a Borel of the space of countable groups.

G In the sequel of this section let Xα := (2) α and Fα := F(Gα, 2). As a consequence of Popa’s superrigidity theorem we easily obtain the following:

Fact 10.26. If α 6= β, then Fα is Fβ-ergodic with respect to the product probability measure on Xα.

An important consequence of the ergodicity result above is that all Fα are not weakly universal. N It follows that ≡T is Fα-m-ergodic, for each α ∈ 2 by Theorem 10.17. Now we are ready to prove Theorem 10.22, which follows from the following result.

Theorem 10.27 (MC). If α 6= β, then (≡T × Fα) is not Borel reducible to (≡T × Fβ).

In proof we shall use a consistency result of Martin and Solovay [33].

ℵ0 1 Theorem 10.28 ([33]). If (MA) + 2 > ℵ1, then every Σ2 set is measurable.

N N Proof of Theorem 10.27. Suppose that f : 2 × Xα → 2 × Xβ is a Borel reduction from (≡T × Fα) N N N to (≡T × Fβ). Let ϕ : 2 × Xα → 2 and ψ : 2 × Xα → Xβ be the Borel function such that N f (r, x) = (ϕ(r, x), ψ(r, x)) for all r ∈ 2 , x ∈ Xα. For each x ∈ Xα, we let

N ψx : 2 → Xβ, r 7→ ψ(r, x).

Notice that r ≡T implies ψx(r) Fβ ψx(s), so ψx gives a Borel homomorphism from ≡T to Fβ for any x ∈ Xα. Since ≡T is Fβ-m-ergodic, for each x ∈ Xα, there is a cone Cx ⊆ Xα such that ψx maps

Cx into a single Fβ-class dx.

Goal. Define a Borel homomorphism from Fα to Fβ and use the fact that Fα is Fβ-ergodic.

51 It is unclear weather such Borel map exists, but fortunately we can work in the more general measure theoretical context and use some metamathematical tricks to get around this difficulty.

First observe that if x Fα y then dx = dy. To see that pick some r ∈ Cy. Notice that if x Fα y, then

(r, x)(≡T × Eα)(r, y), which implies that ψx(r) = ψy(r) ∈ dy. Thus, it suffices to define a function h : Xα → Xβ such that h(x) ∈ dx, for all x ∈ Xα. Consider the following relation:

(x, z) ∈ R ⇐⇒ ∃s ∀r s ≤T r =⇒ ψ(r, x) Fβ z .

1 It is immediate by the definition that R is a Σ2 subset of Xα × Xβ. By a classical result of Kondô [23, Corollary 38.7], there exists a uniformizing function h for 1 R. Observe that whenever r ∈ Cx, we have dx = [ψ(r, x)]Fβ . Thus, the uniformization h is a Σ2 homomorphism from Fα to Fβ. There is no reason to think that such h would be Borel, however if we move to a suitable

P ℵ0 P forcing extention V in which Martin’s axiom (MA) and 2 > ℵ1 hold, then V proves that h is µ-measurable. Hence we can work in VP and use the measure theoretical version of ergodicity

(see Exercise 10.24). Since Fα is Fβ-ergodic, there is Z ⊆ Xα Borel, with µ(Z) = 1, such that h maps

Fβ into a single Fβ-class c. N N Now fix some x ∈ Z, and consider the map ϕx : 2 → 2 , r 7→ ϕ(r, x). It is clear that ϕx is a Borel homomorphism from ≡T to ≡T. Moreover, it cannot be the case that ϕx is constant on a cone. I fact, the restriction ϕx  Cx is a Borel reduction from ≡T  Cx to ≡T. It follows from

Theorem 10.13 (ii) that the saturation [ϕx(Cx)]Fβ contains a cone Dx.

So, if x, y ∈ Z are such that x 6 Fαy, then we can fined some r ∈ Cx and s ∈ Cy such that

ψx(r) ≡T ψx(s). Therefore

f (r, x)(≡T ×Fβ) f (s, y),

P which contradicts with the fact that f is a Borel reduction. To sum up, we have V  ”(≡T × 1 Fα) ≤B (≡T × Fβ)”. That statement is a Π2 property of α and β. So, Shoenfield’s absoluteness theorem implies that (≡T × Fα) ≤B (≡T × Fβ) in the ground model. (See the appendixA.) A crucial point of this proof is that (MC) is equivalent to (MC’), which is therefore absolute. So P 0 another application of Shoenfield’s absoluteness ensures that V  (MC ).

52 A Reductions and absoluteness

Let V be a fixed base universe of set theory and let P be a notion of forcing. Then we will write VP for the corresponding generic extension when we do not wish to specify the generic filter G ⊆ P. P P If R is a projective relation on the Polish space X, then XV , RV will denote the sets obtained by applying the definitions of X, R within VP. In particular, suppose that E is a Borel equivalence relation on the Polish space X. Then the Shoenfield Absoluteness Theorem [?, Theorem 25.20] P P P P implies that XV ∩ V = X and EV ∩ V = E, that EV is a Borel equivalence relation on XV , and that the following result holds.

Theorem A.1. If E, F are Borel equivalence relations on the Polish spaces X, Y and θ : X → Y is a Borel P P P reduction from E to F, then θV is a Borel reduction from EV to FV .

1 Remark A.2. The analogue of the result above for the Σ1 equivalence relations is also true.

53 Concept index

i E vB F, 41 cone, 45 E(G, X), 15 coset equivalence relation, 17 E-m-ergodicity, 48 essentially free equivalence relation, 35 E-invariant, 20

E vB F, 14 free equivalence relation, 34 E ≤ F, 14 B generically ergodic, 12 P(X), 22 α-invariant, 23 hyperfinite, 17 ≤ -cofinal, 46 T lattice, 31 µ-measurable, 11 meager, 11 µ-trivial, 34 measure, 10 action universal, 16 ergodic, 11 amenable, 21 finite, 11 aperiodic,9 non-atomic, 11

Baire measurable, 12 nowhere dense, 11 Baire space,2 orbit equivalence relation,6 Bernoulli shift,8

Borel embeddable, 14 paradoxical, 21 Borel embedding, 14 Polish,2 Borel graphing, 24 Polish group, 17 Borel homomrphism, 14 probability measure, 11 Borel reducibility, 14 push-forward measure, 11 Borel reduction, 14 quasi-automorphism, 38 Borel space,3 quasi-endomorphism, 37 Borel treeing, 24 quasi-equality, 37 Cantor space,2 quasi-isomorphism, 37 cardinal algebra, 41 same type, 28 cocycle property, 25 selector,8 complete section, 17 separating family, 10

54 shift action, 15 smooth,7 standard,3 strongly mixing, 33

Tarkian class, 41 transversal,8 treeable, 24 Turing equivalence, 45 Turing invariant function, 46 Turing reducibility, 45 uniformization,5 uniformizing function,5 universal CBER, 15 virtual embedding, 34 weak Borel reduction, 36 weakly action universal, 49

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