Countable Borel Equivalence Relations

Transcription

1 Countable Borel Equivalence Relations S. Jackson, A.S. Kechris, and A. Louveau This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This study is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras. Classically, in various branches of dynamics one studies actions of the group of integersz, realsr, Lie groups, etc. More generally, one can consider definable (e.g., continuous, Borel, etc.) actions of Polish (i.e., separable, completely metrizable topological) groups on standard Borel spaces (i.e., Polish spaces equipped with their σ-algebra of Borel sets). Most of our emphasis in this paper though will be on Borel actions of Polish locally compact groups. One of the main problems concerning a given definable action of a Polish group G on a standard Borel space X is the complete classification of members of X, up to orbit equivalence, by invariants. (Orbit equivalence being the equivalence relation induced by the orbits of the action.) This is a special case of the more general problem of completely classifying elements Research partially supported by NSF Grants DMS and DMS

2 of a given standard Borel space X up to some definable equivalence relation E defined on that space. This means finding a set of invariants I and a map c : X I such that xey c(x) = c(y), where for this to have any meaning, both I, c must be explicit or definable too. A typical example of this kind of problem is the classification of countable models of a theory up to isomorphism, the classification of the irreducible unitary representations of a Polish locally compact group up to unitary equivalence, the classification of measure preserving transformations up to conjugacy, etc. In measuring the complexity of the classification problem and the nature of the possible complete invariants for a given equivalence relation E, the following notion is important. Let E, E be two equivalence relations on standard Borel spaces X, X. We say that E is Borel reducible to E, in symbols E B E, if there is a Borel map f : X X such that xey f(x)e f(y). Letting then f([x] E ) = [f(x)] E, it is clear that f : X/E X /E is an embedding of X/E into X /E. Intuitively, E B E can be interpreted as meaning any one of the following: (i) E has a simpler classification problem than E : any complete invariants for E work as well for E (after composing with f). (ii) One can classify E-equivalence classes by invariants which are E - equivalence classes. (iii) The quotient space X/E Borel embeds into X /E, so X/E has Borel cardinality less than or equal to that of X /E. Actions of Polish locally compact groups are of course of particular interest in several areas of mathematics. Let G be such a group and consider a Borel action of G on a standard Borel space X. By proving a descriptive set theoretic strengthening of results of Ambrose-Kakutani (see Ambrose [41]) and Feldman-Hahn-Moore [79], it was shown in Kechris [92] that there is a complete countable Borel section for this action, i.e., a Borel set S X which meets every orbit in a countable nonempty set. This essentially reduces the study of the orbit equivalence relation of this action to that of the so-called countable Borel equivalence relations, i.e., Borel equivalence relations all of whose equivalence classes are countable. This subclass of Borel equivalence relations is of great importance in the general theory and its connections with other areas (e.g., ergodic theory), and its study presents some of the deepest and most challenging problems in the whole subject. It is the main topic of our paper. It contains several results of the authors that have been unpublished for several years now (although some of them are included in the widely distributed set of lecture notes Kechris [94a]), as well as results 2

3 obtained by the authors and others more recently, so that it presents an up to date picture of the state of the art in the theory of countable Borel equivalence relations and its main problems. For further reference, let us first introduce some examples of countable Borel equivalence relations. For any standard Borel space X, let (X) be the equality relation on X. By E 0 we denote the following equivalence relation on 2 N : xe 0 y n m n(x(m) = y(m)) (this is sometimes referred to as the Vitali equivalence relation, since it can be viewed as a combinatorial version of the classical Vitali equivalence relation on [0, 1] : xe v y q Q(x + q = y)). For any countable group G and standard Borel space X, let E(G, X) be the equivalence relation on X G induced by the shift action of G on X G : g p(h) = p(g 1 h) for p X G, g G. Finally, T, A are the Turing and arithmetical equivalences on 2 N, resp. In developing the theory of countable Borel equivalence relations, one considers various important subclasses which impose a hierarchy of complexity. The simplest such equivalence relations are those for which E B (2 N ), i.e., those for which there is a Borel function f : X 2 N (where E lives on X) such that xey f(x) = f(y). Thus elements of X can be classified by invariants which are members of 2 N (or in general a standard Borel space). These are called tame (sometimes also concretely classifiable or smooth) Borel equivalence relations and their structure is rather trivial to understand. At the other extreme there is a most complex countable Borel equivalence relation. More precisely, there is a universal countable Borel equivalence relation E, i.e., one that has the property that for every countable Borel equivalence relation E we have E B E. This fact can be shown using the result of Feldman-Moore [77] that the countable Borel equivalence relations are exactly the orbit equivalence relations of Borel actions of countable groups. It is clear that E is unique up to bi-reducibility, B, where E B F E B F & F B E. One particular realization of E is the following: E(F 2, 2), the equivalence relation induced by the shift action of F 2, the free group on 2 generators, on 2 F 2. E is not concretely classifiable. We discuss various examples of universal countable Borel equivalence relations and some interesting open problems about them in Sections 4, 6 below. 3

4 Strictly between the tame and the universal lies the class of hyperfinite Borel equivalence relations. (Every tame relation is hyperfinite but no universal one is.) These are the Borel equivalence relations which are unions E = n E n, with {E n } an increasing sequence of Borel equivalence relations with finite equivalence classes. By a result of Weiss [84] and Slaman-Steel [88], these can be also characterized as the orbit equivalence relations of Z- actions, i.e., single Borel automorphisms. E 0, E v, E(Z, X) are all hyperfinite (but T is not, as proved by Slaman-Steel [88]). Hyperfinite equivalence relations have been studied extensively and classified in Dougherty-Jackson- Kechris [94] both under B and = B (= Borel isomorphism). It turns out that, except for the tame ones, there is, up to B, only one hyperfinite Borel equivalence relation, namely E 0 (which is in particular universal hyperfinite). Thus E 0 B E v B E(Z, 2), etc. Notwithstanding this progress, there are some fundamental problems concerning hyperfiniteness, which, despite the simplicity of their statements, have resisted solution for several years now. These are discussed in Sections 1, 6 below. One of them is the following: (Weiss [84]) Is the orbit equivalence relation induced by a Borel action of a countable amenable group hyperfinite? Recall that a countable group G is amenable if it admits a translation invariant finitely additive Borel probability measure. Extending an unpublished result of Weiss, for the groupsz n, we answer this affirmatively for all finitely generated groups of polynomial growth. In fact, we prove an appropriate generalization of this result for Polish locally compact groups, see Theorem 1.16 below. This is essentially the best known result on this basic problem to date. A wider class than the hyperfinite equivalence relations consists of the socalled amenable ones. These were introduced in this context in Kechris [91] by adapting in the descriptive framework a concept introduced by Zimmer [77] and Connes-Feldman-Weiss [81]. (It was used in Kechris [91] to solve a problem of Slaman-Steel [88] concerning Turing degrees.) The notion of amenability introduced in Kechris [91] involved the concept of universal measurability and was transcending the Borel context. In the present paper a new, more restrictive, notion of amenability is proposed (motivated by some ideas that arose in a conversation with S. Adams and R. Lyons), strictly belonging in the Borel domain, see 2.4 below. It appears to be the correct formulation, and this is partly justified by the results in 2.5 below, but more 4

5 work is needed to verify this. This new concept ramifies the amenable equivalence relations in a non-decreasing hierarchy of ω 1 classes. The hyperfinite equivalence relations belong to the first level and so do all orbit equivalence relations induced by Borel actions of amenable groups. On the other hand, unions of increasing sequences of hyperfinite equivalence relations belong to the second level. It is not known if such unions are hyperfinite (the Union Problem, see Section 6 below). A fundamental problem concerning these notions is whether in fact hyperfiniteness and amenability coincide. An affirmative answer would resolve positively both the Union Problem and Weiss Problem mentioned before. It should be noted here that, by a well-known result of Connes-Feldman-Weiss [81], the answer is affirmative almost everywhere, with respect to any given probability Borel measure. If, on the other hand, it turns out that hyperfiniteness and amenability do not coincide, it is probably quite likely that the aforementioned transfinite hierarchy is proper and provides a true measure of complexity within the concept of amenability. For example, it could happen that some unions of increasing sequences of hyperfinite equivalence relations belong properly in the second level of this hierarchy. Examples of non-amenable equivalence relations include T and the universal one E. The final class of countable equivalence relations that we will consider in this paper are the so-called treeable ones; see Section 3 below. E is called treeable if we can assign, in a uniform Borel way, to each E-class, a tree (connected acyclic graph) with vertices the elements of this class. These include the hyperfinite ones. Other typical examples are those induced by free Borel actions of the free groups on countably many generators. Treeable equivalence relations, in the special case where the trees are locally finite, i.e., have finite degree at each point, were introduced originally in work of Adams [90] in ergodic theory. In the present paper we study the Borel theoretic context and the extension of Adams results to not necessarily l.f. trees. Treeable relations form a strictly wider class than the hyperfinite ones, but E is not treeable. Again it turns out that there is a universal treeable equivalence relation, unique up to B, denoted by E T. It can be realized, for example, as the restriction to the free part (i.e., the points p for which g p p, for g 1) of the orbit equivalence relation E(F 2, 2) induced by the shift action of F 2 on 2 F 2. Thus we have E 0 < B E T < B E (where E < B F E B F & F B E), and this shows that there are 5

6 indeed intermediate (in the sense of B ) countable Borel equivalence relations between E 0 and E. Treeability, in the ergodic theory context, has been also recently studied in Gaboriau [00]. By a combination of work of Gaboriau and the authors it was shown in particular that any treeable equivalence relation E admits a l.f. treeing (in a Borel way), resolving an early open problem in this area; see 3.12 below. On the other hand, Gaboriau [00] has shown that it is not in general possible to find a l.f. treeing in which the degree of every vertex is uniformly bounded. One of the most interesting open questions about treeable equivalence relations is whether there are any which are strictly between E 0 and E T, see Sections 3.5, 6. It was already mentioned that E T is an example of an intermediate, between E 0 and E, equivalence relation, and using work of Adams [88] it can be also shown that E 0 < B E T < B E T E T < B E. So one has at least 4 distinct, up to B, non-smooth countable Borel equivalence relations. For many years now these were the only examples for which one could show that they are different and in particular the following basic problems remained open: Are there infinitely many distinct, up to B, non-tame countable Borel equivalence relations? Are there incomparable under B countable Borel equivalence relations? These problems have been affirmatively solved recently by Adams-Kechris [00], who actually obtained much stronger results. The proofs use, among other things, the machinery of cocycles and the Zimmer Superrigidity Theory (see Zimmer [84]) for ergodic, measure preserving actions of higher rank linear algebraic groups. In analogy with the rigidity phenomena of the ergodic theory of such groups, it is shown that there is a purely descriptive set theoretic rigidity, which, roughly speaking, asserts that simply the Borel cardinality of the orbit space of certain actions of groups remembers or encodes a lot about the acting group. For example, if one considers the canonical action of GL n (Z) ont n, thent n /GL n (Z) has the same Borel cardinality ast m /GL m (Z) iff m = n (invariance of dimension). It is clear that investigating the further implications of this descriptive set theoretic rigidity will be an important part of the theory of countable Borel equivalence relations in the future. We conclude the introduction with some comments on the organization of this paper. After a section (Section 0) of preliminaries in descriptive set theory and the theory of Borel equivalence relations, we study in Section 1 6

7 hyperfinite relations, in Section 2 amenable ones, and in Section 3 treeable ones. In Section 4, we survey universal countable Borel equivalence relations and in Section 5 we discuss some other topics concerning actions of countable groups, like generators and free actions. Finally, in Section 6 we collect together some of the basic open problems in the theory of countable Borel equivalence relations. 0 Preliminaries We will review here some standard terminology and notation concerning the theory of Borel equivalence relations. Some basic background on countable Borel equivalence relations can be found in Dougherty-Jackson-Kechris [94]. For the basic concepts and results in descriptive set theory that we will be using below, see Kechris [95]. A Polish space is a completely metrizable separable topological space. A standard Borel space is a measurable space, i.e., a set X equipped with a σ-algebra S, such that there is a Polish topology on X in which S is its σ-algebra of Borel sets. An equivalence relation E on a standard Borel space X is called Borel if it is a Borel subset of the product space X 2. A Borel equivalence relation E on X is called tame (or smooth or concretely classifiable) if there is a Borel map f : X Y, for some standard Borel space Y, such that xey f(x) = f(y). A selector for E is a map s : X X such that: (i) xey s(x) = s(y), (ii) xes(x). A transversal for E is a subset T X such that T meets every E-equivalence class [x] E of E in exactly one point. It is easy to see that if E is Borel, then E has a Borel selector iff E has a Borel transversal. Moreover, if E has a Borel selector, then E is tame. If E, F are equivalence relations on standard Borel spaces X, Y resp., we say that E is Borel reducible to F, in symbols E B F, if there is a Borel map f : X Y with xey f(x)ff(y). Such an f is called a reduction of E to F. Thus E is tame iff E B (Y ), for some standard Borel space Y, where (Y ) is the equality relation on Y. We also say that E is Borel bireducible to F, in symbols E B F 7

8 if E B F and F B E. Finally, let E < B F E B F & F B E. We say that E is Borel embeddable in F, in symbols E B F if there is a 1-1 Borel reduction of E to F. Let also E B F E B F and F B E. If the range of such a 1-1 Borel reduction is actually a (necessarily Borel) F-invariant subset of Y we write E i B F. Note that by a standard Schroeder-Bernstein argument: E i B F & F i B E E = B F, where = B denotes Borel isomorphism, i.e., the existence of a Borel bijection between X and Y such that xey f(x)f f(y). A Polish group is a topological group whose topology is Polish. If X is a standard Borel space, and G is a Polish group, a Borel action of G on X is an action (g, x) g x of G on X which is a Borel map from G X into X. We also refer to the triple (G, X, ) as a (standard) Borel G-space. For an introduction to the theory of Borel G-spaces, see Becker-Kechris [96]. If X is a Borel G-space, we denote by E X G the orbit equivalence relation induced by the action: xe X G y g G(g x = y). If X is understood or irrelevant, we simply write E G instead of E X G. For example, E Z denotes an equivalence relation induced by a Borel action of the groupzon some standard Borel space, i.e., one given by the orbits of a single Borel automorphism. An action of G on X is free if for any g 1 (= the identity of G) and any x X, g x x. If X is a Borel G-space and the action is free, E X G is Borel. (In general it is analytic.) 8

9 A Borel equivalence relation E on a standard Borel space X is countable if every equivalence class [x] E is countable. For example, if G is a countable (discrete) group and X is a Borel G-space, then EG X is a countable Borel equivalence relation. Conversely, Feldman-Moore [77] showed that every countable Borel equivalence relation E is of the form E G, for some countable group G. For each countable group G and standard Borel space X, consider the shift action of G on the (standard Borel) space X G given by g p(h) = p(g 1 h), for g G, p X G. We denote by E(G, X) the corresponding equivalence relation. It is of course a countable Borel equivalence relation. It is not hard to check that if G F (i.e., G is a subgroup of F), then E(G, X) B E(F, X). If F n is the free group with n generators, then E(F 2, 2) has the following universality property: if E is a countable Borel equivalence relation, then E B E(F 2, 2). Finally, we denote by F(G, X) the restriction of E(G, X) to the free part of the shift action, i.e., the (Borel G-invariant) set {p X G : g G(g 1 g p p}. Finally, we recall some basic definitions from ergodic theory. If E is a countable Borel equivalence relation on a standard Borel space X and µ is a Borel probability measure on X, then we call the measure µ E-invariant if for any countable group G and Borel action of G on X with E = E X G, we have that µ is G-invariant, i.e., g µ = µ, g G (where g µ(a) = µ(g 1 A), for A any Borel set). We call µ E-ergodic if for every E-invariant Borel set A X, we have µ(a) = 0 or µ(a) = 1. The measure µ is E-quasiinvariant if for every G as above the measure µ is G-quasi-invariant, i.e., g µ µ, g G (where for measures µ, ν we have µ ν iff Borel set A(µ(A) = 0 ν(a) = 0)). A countable Borel equivalence relation E on a standard Borel space X is compressible if there is a Borel 1-1 map f : X X such that for each x X, f([x] E ) is a proper subset of [x] E. By a theorem of Nadkarni [90], E is compressible iff there is no E-invariant probability Borel measure. 9

10 1 Hyperfinite Equivalence Relations 1.1 Basic facts and examples Definition 1.1 A Borel equivalence relation E on a standard Borel space X is finite if every equivalence class [x] E of E is finite. It is called hyperfinite if E = n E n, where (E n ) is an increasing sequence of finite Borel equivalence relations. Note that in the previous definition, the fact that the sequence is increasing is crucial, for it can be seen that any countable Borel E can be written as a union of finite Borel E n s, even with E n s having classes of cardinality at most 2 (see Feldman-Moore [77]). Hyperfinite equivalence relations have been extensively studied, see Dougherty-Jackson-Kechris [94] and references therein. Here we review some basic facts about them. First, some equivalent reformulations of hyperfiniteness. Proposition 1.2 (Weiss, Slaman-Steel, Dougherty-Jackson-Kechris) The following are equivalent, for E a countable Borel equivalence relation on a standard Borel space X: i) E is hyperfinite. ii) E = n E n, where the E n s are increasing finite Borel equivalence relations, and all E n -classes have cardinality at most n. iii) E = n E n, where the E n s are increasing tame Borel equivalence relations. iv) E = E Z, i.e., there is a Borel automorphism T of X such that xey n Z (y = T n x). v) There is a Borel partial order on X which induces on each E-class a linear order of typezor finite (or equivalently there is a Borel assignment C < C giving for each equivalence class C a linear order < C on C of order typezor finite the assignment being Borel meaning that R(x, y, z) x < [z]e y is Borel). The following are some basic closure properties of hyperfiniteness: Proposition 1.3 Let E, F, (E i ) i N be countable Borel equivalence relations on X, Y, (X i ) i N respectively. (i) If X = Y, E F and F is hyperfinite, so is E. (ii) If F is hyperfinite and E B F, then E is hyperfinite. 10

11 (iii) If E is hyperfinite and A X is Borel, E A is hyperfinite. (iv) If E, F are hyperfinite, so is E F on X Y (defined by (x, y)e F(x, y ) xex and yfy ). (v) If each E i is hyperfinite, i E i is hyperfinite (defined on i X i = i {i} X i by (i, x) i E i(j, y) i = j and xe i y). (vi) Suppose A is a Borel complete section of E, i.e., a Borel subset of X which meets all E-equivalence classes. Then if E A is hyperfinite, so is E. (vii) If X = Y, E F, E is hyperfinite and every F-equivalence class contains only finitely many E-classes, then F is hyperfinite. Proof. (i), (iii), (iv), and (v) are obvious. (ii) follows from (vi), for if f : X Y is Borel with E = f 1 (F), f is countable-to-1, so B = f(x) is Borel and f admits a Borel inverse g : B X. Let A = g(b). Then A is a Borel complete section of E, and E A = B F B, so by iii) E A is hyperfinite. (vi) Let G = {g n : n N} be a countable group with E X G = E. For x X, let N(x) = least n(g n x A). Clearly N is Borel. If now E A = n F n, with F n finite increasing Borel equivalence relations, set xe n y x = y or (N(x), N(y) n and g N(x) xf n g N(y) y). Then the E n s are increasing finite Borel equivalence relations, and E = n E n. vii) Since for 1 k, the set X k = {x : [x] F contains exactly k E-classes} is Borel, we may assume that each F-class contains exactly k E-classes. Let G = {g n : n N} be a countable group with E X G = F, and inductively define f 1 (x),, f k (x) by f 1 (x) = x, and if f 1 (x),, f i (x) have been defined, let and N i+1 (x) = least n( g n xef j (x), for j i), f i+1 (x) = g Ni+1 (x) x Then the f i s are Borel, and [x] F = k i=1 [f i(x)] E. Let E = n E n, with E n increasing finite Borel equivalence relations. Put then xf n y there is a permutation π of {1,, k} such that i {1,, k}f π(i) (x)e n f i (y). 11

12 This gives an increasing sequence of finite Borel equivalence relations with union F. Examples 1.4 (A) Define E 0 on the Cantor space 2 N by Then E 0 is hyperfinite. xe 0 y n m n(x(m) = y(m)). Note that viewing 2 N as a group, with coordinate-wise addition mod 2, E 0 is the coset-equivalence relation associated with the sub-group 2 <N = n N 2n. More generally, if G is a Polish group, n Nand H = n H n is a subgroup, where all H n s are countable discrete subgroups of G and the sequence is increasing, then the corresponding (left) coset equivalence relation E G H given by ge G Hh h gh is hyperfinite. This applies in particular to the Vitali equivalence relation E V onr given by xe V y y x Q, asq = n (n!) 1 Z. (B) The tail equivalence relation E t on 2 N is hyperfinite, where xe t y n m k(x(n + k) = y(m + k)). This example as well as E 0 can be generalized: Let X be a Polish space, U a Borel countable-to-one function from X to X. Define xe 0 (U)y n(u n x = U n y), xe t (U)y n m(u n x = U m y). Then E 0 (U) and E t (U) are hyperfinite (Dougherty-Jackson-Kechris [94]). (E 0 and E t correspond to the case where U is the one-sided shift on 2 N.) 12

13 ( ) a b (C) Consider the group GL 2 (Z) of all matrices with integer coefficients ( and ) determinant ad bc = ±1. Let GL 2 (Z) act onr { } by c d a b x = c d ax+b. Then the orbit equivalence relation induced by cx+d this action can be described as follows (see Hardy and Wright [68]): All elements of Q { } form one equivalence class. For irrationals α, β let α = [a 0, a 1, ], β = [b 0, b 1, ] be their continued fraction expansions. Then α, β are equivalent iff (a n )E t (Z)(b n ), where E t (Z) = E t (U), with U :Z N Z N the one-sided shift. Thus this is a hyperfinite equivalence relation. (D) E(Z, 2), defined on 2 Z by using the canonical action ofzon 2 Z by shift, is clearly hyperfinite by 1.2. By general results it is universal for all orbit equivalence relations induced by Z-actions, i.e., any hyperfinite E is Borel-reducible to E(Z, 2). (E) Consider the free group F 2 with two generators a, b and let W be the set of all infinite reduced words in {a, b}, i.e., infinite sequences (x i ) such that each x i is one of a, b, a 1, b 1 and x i x i+1 1. The group F 2 acts on W in the obvious way by left concatenation and cancellation. It is then easy to see that (x i ), (y i ) are in the equivalence relation induced by this action iff (x i )E t (U)(y i ), where U is the (one-sided) shift on {a, b, a 1, b 1 }. So this equivalence relation is hyperfinite. But this last fact is indeed true of all non-tame hyperfinite equivalence relations, by the following result of Dougherty-Jackson-Kechris, which completely classifies them up to bi-reducibility. Theorem 1.5 (i) (Glimm-Effros) If E is a countable Borel equivalence relation which is non-tame, then E 0 B E. (ii) (Dougherty-Jackson-Kechris) If moreover E is hyperfinite, E 0 B E. So the hyperfinite non-tame equivalence relations form one B -degree which is minimum in the B ordering on countable Borel equivalence relations above the tame ones. The following result completely classifies up to Borel isomorphism the hyperfinite equivalence relations which are non-tame and aperiodic (i.e., without finite classes). The general case follows easily, using the easy classification of tame equivalence relations, hence of the periodic part of hyperfinite equivalence relations. 13

14 Theorem 1.6 (Dougherty-Jackson-Kechris) Let E be a hyperfinite non-tame and aperiodic Borel equivalence relation. Then E is Borel-isomorphic to exactly one of the following: E t, E 0, E 0 (2), E 0 (3), E 0 (n),, E 0 (N), E 0 (2 N ) (or the aperiodic part of E(Z, 2)), where for each X, (X) is equality on X. Recall that a probability Borel measure µ on X is said to be E-invariant, where E is some countable Borel equivalence relation on X, if for some (any) countable group G and Borel action of G on X with E = EG X, µ is invariant under the action of G. And µ is E-ergodic if every E-invariant Borel set has µ-measure 0 or 1. The proof of the previous theorem goes by showing that for hyperfinite, non-tame and aperiodic E on X, the cardinality of the set E(E) of E-invariant E-ergodic probability measures on X completely characterizes E up to Borel isomorphism. Noting that E t has no such invariant ergodic measures, and that E(E 0 (X)) has cardinality that of X gives the above result. It seems to follow from the two previous results that hyperfinite equivalence relations are well-understood. This is true in some sense, but the main remaining problem is that one does not have a concrete practical way of deciding, for a given countable E, whether it is hyperfinite or not. As a consequence, many basic natural questions about hyperfinite equivalence relations are still open (see section 6). 1.2 Which groups give hyperfinite actions? We have seen that every E Z is hyperfinite. How about E G, for more general countable groups G? We will see later that all E G are hyperfinite, when G is finitely generated abelian (Weiss) and more generally if it is finitely generated of polynomial growth. On the other hand, E(F 2, 2) is not hyperfinite. This follows from the next general fact. Recall that a countable group G is amenable if there is a (left) translationinvariant finitely additive probability measure (f.a.p.) ν on G, or equivalently a positive element ν l (G), invariant under the (left) shift on l (G), and with ν(1) = 1 (such a ν is called an invariant mean on G). Proposition 1.7 Let G be a countable group, and (g, x) g x a Borel action of G on the standard Borel space X. Let µ be a G-invariant probability Borel 14

15 measure on X and assume that on a µ-measure 1 set X the G-action is free, i.e., x X g 1(g x x). Then if EG X is hyperfinite, G is amenable. (We will prove this in 2.5(ii).) Corollary 1.8 If G is a countable group, and all Borel actions of G are hyperfinite, in fact even if just E(G, 2) is hyperfinite, then G is amenable. In particular E(F 2, 2) is not hyperfinite. Proof. Assume E = E(G, 2) is hyperfinite. Let λ be the usual coin flipping probability measure on 2 G. Easily, λ is E-invariant, and is supported by the free part of the action of G on 2 G. So by proposition 1.7, G must be amenable. For the last fact, just note that F 2 is not amenable. It is not known whether the converse to proposition 1.7 holds, i.e., whether every countable amenable group G has all its orbit equivalence relations hyperfinite. This problem has been raised in Weiss [84]. We will establish some partial results in this direction, but before that, we will discuss some more examples and an extension of the concept of hyperfiniteness. Definition 1.9 Let E be a Borel equivalence relation and F a countable Borel equivalence relation. We say that (i) E is essentially F if E B F. (ii) E is essentially countable if for some countable Borel H, E B H. (iii) E is essentially hyperfinite if for some Borel hyperfinite H, E is essentially H. Remark. It is not known whether, in general, E is essentially countable iff E is essentially F for some countable F. It holds though in the special case when E is induced by a Borel action of a Polish group. Examples 1.10 (A) Polish locally compact group actions Theorem 1.11 (Kechris [92]) Let G be a Polish locally compact group, and EG X the orbit equivalence relation associated to some Borel action of G on a standard Borel space X. Then EG X is essentially countable; in fact there exists a Borel complete section A of E such that EG X A is countable, so that in particular EG X B EG X A. It is not known if there is a converse to 1.11, i.e., if G is a Polish group with all E G s essentially countable, is G necessarily locally compact? Solecki 15

16 [99] has proved that if all E G s are tame, G is compact. Some further partial results on this problem can be found in Kechris [92]. Remark. We take this opportunity to point out that the question raised in the first paragraph of p. 287 in Kechris [92] has a negative answer. The infinite symmetric group S of all permutations onnadmits an a.e. free Borel action on [0,1] N with invariant probability measure (the usual product measure), but S cannot be a Borel subgroup of a Polish locally compact group. Another question related to 1.11 is the existence of a countable companion for a given Polish locally compact group G, i.e., some countable group G with the property that up to B the E G are the same as the E G. We will see in a moment thatrand ther n s, n N, admitzas a countable companion. But this is a very particular case. For the general case, the proof of 1.11 falls short of this. It only gives the existence of a lacunary complete Borel section A, i.e., some complete Borel section with the property that for some neighborhood V of 1 one has x A (V x A = {x}). (B) Actions of closed subgroups of S Theorem 1.12 (Hjorth-Kechris [96], Hjorth-Kechris-Louveau [98]). Let G be a closed subgroup of S, and EG X the orbit equivalence relation induced by a Borel action of G on some standard Borel space X. Then the following are equivalent: (i) EG X is essentially countable, (ii) For some Polish topology τ on X giving its Borel structure, EG X is Σ0 2 in (X, τ) 2 ; (iii) For some Polish topology τ on X giving its Borel structure, EG X is Σ 0 3 in (X, τ) 2. Note that 1.12 cannot be extended to arbitrary Polish groups. For example, if X =R N, G = l 2 and E G is the coset equivalence (x n )E G (y n ) (x n y n ) n N l 2, then E G is a Σ 0 2 non essentially countable equivalence relation. Also Σ 0 3 is best possible. If E is the equivalence relation on [R] N = {x R N : x is 1 1} given by xey {x n : n ω} = {y n : n ω}, then E is of the form E S, it is Π 0 3, but is not essentially countable. It is known that equivalence relations of the form E G, G a closed subgroup of S, are B to the isomorphism relation on the space of countable models of some L ω1 ω sentence σ in some countable language (Becker-Kechris [96]). 16

17 So let us say that σ (or the L ω1 ω-theory T whose axiom is σ) is essentially countable if (Mod(σ), =) is. Here are some examples of such theories (see Hjorth-Kechris [96] for more such examples, and for model-theoretic versions of 1.12). the theory of finitely generated groups (and more generally of finitely generated structures in any countable language); the theory of finite rank torsion-free abelian groups; the theory of fields of finite transcendence degree overq; the theory of locally finite (i.e., those for which every vertex has finite degree) trees; the theory of locally finite connected graphs. It is known that the theory of rank 1 torsion free abelian groups is essentially hyperfinite. The theories of locally finite trees and locally finite connected graphs are essentially E(F 2, 2) (see section 4), as is the theory of finitely generated groups, by a result of Thomas and Velickovic [99]. For torsion-free abelian groups of any given finite rank 2, it was recently shown by Thomas that it is strictly below E(F 2, 2). Concerning the relationship between essential countability and essential hyperfiniteness, one has the following result, based on earlier work of Sullivan- Weiss-Wright [86] and Woodin: Theorem 1.13 (Hjorth-Kechris [96]) Let X be a Polish space, and E an essentially countable Borel equivalence relation on X. Then there exists an E-invariant comeager Borel set C X such that E C is essentially hyperfinite. So from the point of view of category, the two notions coincide. Note that by the discussion in corollary 1.8, no such result can hold from the measuretheoretic point of view, as E(F 2, 2) and λ on 2 F 2 exemplify. We will see in the next section in what cases one can get essential hyperfiniteness on a measure 1 set. We now turn to the question of which groups G have the property that all associated E G s are essentially hyperfinite. Weiss (unpublished) has proved that all free Borel actions ofz n, n N, have hyperfinite orbit equivalence 17

18 relations. Building on his work, we will prove here that Polish locally compact groups G which are compactly generated of polynomial growth have essentially hyperfinite E G s. As our proof gives a slightly better result, we first introduce some definitions. Definition 1.14 Let G be a Polish locally compact group, µ G its right Haar measure. (i) G is compactly generated of polynomial growth d if there exists a symmetric compact nbhd K of 1 which generates G (G = n Kn ) and an integer d with µ G (K n ) O(n d ). (ii) Let (K n ) be an increasing sequence of symmetric compact sets in G. We say that (K n ) has mild growth of order c > 0 if for infinitely many n s, µ G (K n+4 ) cµ G (K n ). (iii) We say that G has the mild growth property of order c if there exists an increasing sequence (K n ) in G of mild growth of order c such that K 0 is a nbhd of 1, K 2 n K n+1, and n K n = G. Proposition 1.15 (i) Let G be a compactly generated Polish locally compact group of polynomial growth d. Then G has the mild growth property of order c = 16 d + 1. (ii) Suppose G is a Polish group which is an increasing union of Polish locally compact groups (G i ) which have the mild growth property of order c. Then G has the mild growth property of order c. Proof. (i) Let K be a generator of G with µ G (K n ) An d for some A > 0. It is clearly enough to check that K n = K 2n form a sequence of mild growth c = 16 d + 1. Otherwise, there is n 0 such that for n n 0, µ G (K n+4 ) cµ G (K n ), so that µ G (K 4k+n0 ) c k µ G (K n0 ) for all k. On the other hand, µ G (K 4k+n0 ) A 2 dn0 2 4dk, and as µ G (K n0 ) > 0, this can happen only if c 2 4d = 16 d. This proves (i). (ii) Recall that if G is Polish and H G is a non meager Borel subgroup, H is clopen in G. So, by the Baire category theorem, we can assume that the G i s are clopen subgroups of G. In particular, G is locally compact and µ G G i = µ Gi. Fix a generating sequence K n (i) of mild growth c for each G i, and a bijection ϕ = (ϕ 0, ϕ 1 ) :N N N, with the property that ϕ 0 (k) k. Define inductively a sequence (K n ) as follows. Suppose the K i s have been defined for i < 5k, with K i G i. Let K = K5k 1 2 K(ϕ 0(k)) ϕ 1 (k). Then K G 5k is compact. Moreover, as 1 K (5k) 0, K n (5k) Int(K n (5k) ) K n (5k) Int(K (5k) n+1 ), hence 18

19 G 5k = n Int(K(5k) n ). So, by compactness, one can find n large enough so that K K n (5k) and µ G (K n+4) (5k) cµ G (K n (5k) ). Set for i = 0,, 4, K 5k+i = K (5k) n+i. By the construction, (K n) has mild growth c, K 0 is a nbhd of 1, Kn 2 K n+1 for all n and n K n p,i K(i) p = G. In case G is countable, being compactly generated of polynomial growth just means being finitely generated of polynomial growth, and, by a famous result of Gromov, this is equivalent to being finitely generated nilpotent-byfinite. So finitely generated nilpotent-by-finite groups have the mild growth property by the previous proposition. But this proposition can be applied to show that other (non necessarily finitely generated) countable groups also have this property, likeq or theq n s (write againq= n (n!) 1 Z, and similarly for theq n s). Theorem 1.16 Let G be Polish locally compact with the mild growth property, and EG X be the orbit equivalence relation induced by a Borel G-action on a standard Borel space X. Then EG X is essentially hyperfinite. We first establish a series of lemmas. Let F be a symmetric and reflexive Borel relation on the standard Borel space X. F is locally finite if for all x X, F(x) = {y X : yfx} is finite. A subset Y of X is F-discrete if x, y Y (x y xfy), and is maximal F-discrete if it is F-discrete, and moreover x X y Y (xf y). Lemma 1.17 Let F be a locally finite symmetric reflexive Borel relation on X. Then there exists a Borel maximal F-discrete subset of X. Proof. Let (B n ) n N be a sequence of Borel subsets of X which separates points and is closed under finite intersections, and for x X set ϕ(x) = inf{n : F(x) B n = {x}}. This is clearly well-defined as F(x) is finite, and ϕ is Borel. Moreover, for every n, ϕ 1 (n) is a Borel F-discrete set. Define then inductively, Y 0 = ϕ 1 (0), and Y n+1 = ϕ 1 (n + 1)\ F(y). j n y Y j Again the Y n s are Borel (as F is locally finite), and one checks that Y = n Y n is maximal F-discrete. Definition 1.18 Let (F n ) n N be a sequence of locally finite symmetric reflexive Borel relations on X. We say that (F n ) satisfies the Weiss condition if Fn 2 F n+1 for all n, and there is a constant c > 0 such that 19

20 (*) x X there are infinitely many n such that any F n -discrete set contained in F n+2 (x) has cardinality c. Lemma 1.19 Suppose (F n ) n N is a sequence of locally finite symmetric reflexive Borel relations on X satisfying the Weiss condition. Let E = n F n. Then E is a hyperfinite Borel equivalence relation. Proof. By lemma 1.17, let Y n X be maximal F n -discrete and Borel, and let π n : X X be Borel with π n (x) F n (x) Y n. Define π n = π n π n 1 π 0, and set xe n y π n (x) = π n (y). Clearly each E n is a finite Borel equivalence relation, and E n E n+1, so that E = n E n is hyperfinite. One has E E, and in view of 1.3.(vii), it is enough to prove that each E-equivalence class contains at most c E - equivalent classes. Suppose not, and let x 0, x 1,, x N, N c, be E-equivalent but not E - equivalent elements. Fix n large enough so that i N(x i F n (x 0 )), and (by the Weiss condition) any F n -discrete subset of F n+2 (x 0 ) has cardinality c. Let then for i = 0, 1,, N, y i = π n (x i ). Then the y i s are distinct elements of Y n, hence form an F n -discrete set. Moreover y i F n+1 (x i ), hence y i F n+2 (x 0 ). This contradiction finishes the proof. Proof of 1.16 Pick (K n ) and c > 0 so that (K n ) has the mild growth property of order c, 1 Int(K 0 ), K 2 n K n+1 and n K n = G. Let (g, x) g x be the Borel G-action on X. By Becker-Kechris [96], there is a Polish topology on X giving its Borel structure such that the action is continuous. By the proof of theorem 1.11 (see the third paragraph following 1.11), we can assume that E G admits a Borel complete section Y which is K 1 -lacunary, i.e., such that y Y (K 1 y Y = {y}). E G Y = E is Borel countable, and easily E G B E, so it is enough to prove that E is hyperfinite. Define, for x, y Y, xf n y g K n (g x = y). Clearly each F n is Borel, reflexive, symmetric, F 2 n F n+1 and n F n = E. So by lemma 1.19, it is enough to check that F n is locally finite, and has the right growth property. Pick x Y. We first check that F n (x) is finite. For each y F n (x), pick g(y) K n with g(y) x = y. Now note that for y, z F n (x) with y z, one has K 0 g(y) K 0 g(z) =. For, otherwise, one gets p, h K 0 with pg(y) = hg(z), hence in particular h 1 p g(y) = g(z). As h 1 p K 1, this contradicts the K 1 -lacunarity of Y. So the family (K 0 g(y)) y Fn(x) is a family 20

21 of disjoint compact sets of Haar-measure µ G (K 0 ) > 0, and all contained in K 0 K n K n+1. As µ G (K n+1 ) <, F n (x) must be finite. We now check the Weiss property. Fix again x Y, and let n be such that µ G (K n+4 )/µ G (K n ) c. Suppose {x 1,, x N } is F n+1 -discrete in F n+3 (x). Let g 1,, g N K n+3 be such that g i x = x i, for i N. By the same argument as before one has K n g i K n g j = for 1 i j N. As K n g i K n+4 for all i, we get µ G (K n+4 ) Nµ G (K n ). So N c and we are done. Corollary 1.20 If G is finitely generated nilpotent-by-finite, or if G =Q n, all Borel actions of G induce hyperfinite orbit equivalent relations. We do not know if our technique of proof can be pushed further. In particular we do not know if the corollary still holds when G is countable abelian, or finitely generated solvable. For countable abelian groups the problem is equivalent to that for the groupz <N, the direct sum of infinitely many copies of Z. Using some facts from Dougherty-Jackson-Kechris [94], it is not hard to see that this is further equivalent to the question of whether the equivalence relation induced by the action ofz <N on p(z <N )(= the power set ofz <N ) by translation is hyperfinite. The groupz <N is isomorphic to the multiplicative group (Q +, ) of positive rationals. If we viewq + as acting on p(q + ) by multiplication, then we get an equivalence relation Borel isomorphic to the preceding one and our problem is of course equivalent to whether this is hyperfinite. Finally, by identifying reals with Dedekind cuts, we can viewr + = {r R:r>0} as an invariant, for this action, Borel subset of p(q +) and the restriction of our equivalence relation to it is simply the commensurability relation on positive reals: For x, y R +, xey x/y Q. This is perhaps the simplest equivalence relation for which it is still open whether it is hyperfinite or not. We conclude this section with an amusing fact. For a pair of equivalence relations E, F on X we denote by E F the smallest equivalence relation containing E, F. We say that an equivalence relation E has type n if every E-class has cardinality n. Proposition 1.21 (i) For every countable Borel equivalence relation E, there is a countable Borel equivalence relation F with E B F such that F = G H, where G, H, are Borel of types 2,3 respectively. 21

22 (ii) The equivalence relations of the form E F with E, F Borel of type 2 are exactly the hyperfinite ones. Proof. (i) Let I(N) =N N. Then E B E I(N) and E I(N) is compressible (see Dougherty-Jackson-Kechris [94], 2.5) so we can assume that E is compressible. Let K = PSL 2 (Z) = a, b; a 2 = b 3 = 1. Then F 2 K, so E(F 2, 2) B E(K, 2), thus, as E B E(F 2, 2), we have E B E(K, 2) and, as E is compressible, E i B E(K, 2) (see Dougherty-Jackson-Kechris [94], 2.4), so E is induced by an action of K, thus easily E = G H, where G, H are of types 2,3 respectively. (ii) If R = E F with E, F Borel of type 2, clearly R is induced by a Borel action of G = a, b; a 2 = b 2 = 1, which has polynomial growth, so by 1.5(ii), R is hyperfinite. Conversely, assume R is Borel hyperfinite. We can assume that R is non-tame and aperiodic (i.e., has no finite classes). Let then S be an equivalence relation induced by a Borel action of G with 2 ℵ 0 invariant ergodic probability measures. By Dougherty-Jackson-Kechris [94], , R i B S, so R is induced by a Borel action of G, thus R = E F with E, F of rank 2. Remark. V. Kanovei has pointed out that one also can give a direct proof of 1.21 (ii) above. This leaves open the question of whether every countable Borel equivalence relation E can be written as E = G H, for G, H Borel of types 2,3 resp. In an earlier draft of this paper, we conjectured that this failed and in fact that there are countable Borel E which cannot be written as E = H 1 H n, for any finite Borel H 1,..., H n. It turns out that this is indeed the case, as it follows immediately from recent results of Gaboriau [00]. 2 Amenable Equivalence Relations 2.1 Amenable groups Given a countable set C, a finitely additive probability measure (f.a.p.) on C is a function ϕ: Power(C) [0, 1] such that ϕ is additive, i.e., ϕ(a B) = ϕ(a) + ϕ(b), when A B =, and ϕ(c) = 1. A mean on C is a positive linear functional ϕ on l (C), the Banach space of bounded real functions on C, with ϕ(1) = 1 (thus ϕ is also continuous). Means and f.a.p. s are 22

23 essentially the same thing: Given ϕ, define ϕ by ϕ(f) = fdϕ. Conversely given ϕ, define ϕ(a) = ϕ(1 A ). We will usually identify ϕ and ϕ as above in the sequel. Recall that a countable group G is amenable if there is a mean ϕ on G which is left invariant. How simple can such a ϕ be? By a result of Pinkus and Solovay any f.a.p. ϕ on a countable set C with ϕ({c}) = 0 for all c C is not Baire measurable. So unless G is finite, we cannot hope for a simply definable, e.g., Borel, such ϕ. The best one can hope for are measurability properties. We will use the following result proved independently by Christensen [74] and Mokobodzki (see Dellacherie-Meyer [83]). Theorem 2.1 (Christensen [74], Mokobodzki) (i) Let µ be a Borel probability measure on [ 1, 1] N. There is a mean ϕ µ onnsuch that (a) ϕ µ [ 1, 1] N is µ-measurable, and (b) ϕ µ is non-atomic: ϕ µ (f) = 0, if f is eventually 0. (ii) Assuming the Continuum Hypothesis, CH, one can find a non-atomic mean ϕ on N which is universally measurable. Notice that if ϕ µ is as in 2.1, and we define ψ µ by ψ µ (f) = ϕ µ (n f(0) + f(1) + + f(n 1) ), n then ψ µ is moreover shift-invariant, i.e., ψµ (f) = ψ µ (f s ), where f s (n) = f(n + 1). Similarly for ϕ. In terms of the associated f.a.p., this means that for every Borel probability measure µ on 2 N one can find a shift-invariant f.a.p. ϕ onnwhich is µ-measurable, and under CH, such a ϕ which is universally measurable. Now suppose that X is a standard Borel space, and (f n ) a bounded sequence of Borel real functions on X. If ϕ is a universally measurable shiftinvariant mean onn, the function f(x) = ϕ(n f n (x)) on X is universally measurable. Mokobodzki proves that under CH one can choose ϕ with the additional property that for all X, (f n ), f as above and µ a Borel probability measure on X, one has fdµ = ϕ(n f n dµ). Such ϕ is called a medial measure. We will not need this improvement here. 23

24 The hypothesis CH is not really necessary to get the existence of universally measurable shift-invariant means onn. Martin s Axiom, or even some weakenings of it, suffices. However it is not known if 2.1 (ii) can be proved in ZFC alone. Corollary 2.2 Let G be a countable amenable group. For every Borel probability measure µ on [ 1, 1] G, there is a µ-measurable left-invariant mean on G. Moreover, assuming CH, there is a universally measurable left-invariant mean on G. Proof. Recall that when G is an amenable countable group, there exists a sequence (F n ) of finite nonempty subsets of G (a Fölner sequence) such that for all g G, card(gf n F n ) 0, when n. card(f n ) Let p : l (G) l (N) be given by p(f)(n) = (card(f n )) 1 g F n f(g). Let ν = pµ and let ϕ ν be as in 2.1 (i). One easily checks that ϕ(f) = ϕ ν (p(f)) works. Similarly with ϕ given by 2.1 (ii) for the second part of the result. Remark. In 2.2, we can take the means to be right-invariant, and in fact invariant. To see this, notice that if ϕ l, ϕ r are resp. left, right invariant means, the ϕ defined by ϕ(f) = ϕ r (g ϕ l (f g )), where f g (h) = f(hg), is invariant. 2.2 µ-amenability for equivalence relations Now let E be a countable Borel equivalence relation on a standard Borel space X. There are different possible definitions for E being amenable, and we now discuss some of them. First we deal with the measure-theoretic context. Let µ be a Borel probability measure on X. The notion of µ-amenability for E has been originally defined by Zimmer [77]. Let B be a separable real Banach space, LI(B) the group of its linear isometries, which is Polish under the strong operator topology. Let 24