EQUIVALENCE RELATIONS WITH AMENABLE

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES NEED NOT BE AMENABLE Vadim A. Kaimanovich

Dedicated to the memory of Vladimir Abramovich Rokhlin. Abstract. There are two notions of amenability for a countableequivalencerelation on a Lebesgue space. One (\global") is the property of existence of leafwise invariant means, whereas the other (\local") applies to equivalence relations endowed with an additional leafwise graph structure and means that a.e. leafwise graph has subsets with arbitrary small isoperimetric ratio. In the present article we show that local amenability does not imply global amenability contrary to a widespread opinion expressed in a number of earlier papers. We also formulate a general necessary and sucient condition of global amenabilityin terms of leafwise isoperimetric properties.

There are two notions of amenability for a countable equivalence relation R on a Lebesgue space (X; ). The \global" amenability (which is usually referred to just as \amenability") is the property of existence of leafwise invariant means, which, by a theorem of Connes{Feldman{Weiss [CFW81], is equivalent to hyper niteness, or, to being the orbit equivalence relation of a Z-action. In a way, the origin of this concept can be traced back to the famous Rokhlin{Halmos approximation lemma on Z-actions with a nite invariant measure (e.g., see [Se79]). The notion of \local" amenability applies to equivalence relations endowed with an additional leafwise graph structure and means that a.e. leafwise graph is amenable (or, Flner) in the sense that it has subsets A with arbitrary small isoperimetric ratio j@Aj=jAj (equivalently, that 0 belongs to the spectrum of leafwise Laplacians). In the present article we exhibit examples showing that local amenability does not imply global amenability contrary to a widespread opinion expressed in a number of earlier papers. We construct these examples both in the measure-theoretical (for countable equivalence relations) and in the continuous (for foliations) categories (Theorems 1 and 3, respectively). We also formulate a general necessary and suf cient condition of global amenability in terms of leafwise isoperimetric properties (Theorem 2), which demonstrates that there are only two obstacles for equivalence of global and local amenability: possible non-invariance of the measure  and the fact that amenability of a graph is not necessarily inherited when passing to a subgraph. Theorem 2 shows how one has to modify the leafwise Flner condition in order to take care of both these obstacles. 1991 Mathematics Subject Classi cation. Primary 28D99, 43A07, 53C12, 57R30; Secondary 05C75, 58G25. Key words and phrases. Amenability, equivalence relation, invariant mean, Flner condition. A part of this work was done during the author's stay at the University of Manchester whose support is gratefully acknowledged. Typeset by AMS-TEX 1

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VADIM A. KAIMANOVICH

However, the question whether local amenability implies global amenability for foliations of compact manifolds with a nite transverse invariant measure apparently remains open. By Theorems 1 and 2 this question can not be answered positively by using only the reduction from the continuous to the measure-theoretical category without taking into account the compactness condition. 1. Amenability

We begin with recalling the de nition of amenable groups. Denote by l11(G) the space of probability measures on a countable group G, and by (l1 )1 (G) the space of normalized positive linear functionals on l1 (G), i.e., the space of means ( nitely additive probability measures) on G. Obviously, niteness of G is equivalent to existence of a nite invariant measure on G:

9 m 2 l11 (G) : gm = m 8 g 2 G :

(1)

There are two natural ways of generalizing property (1): either to look for xed points in the larger space (l1 )1 (G)  l11 (G), which gives the condition

9 p 2 (l1 )1 (G) : gp = p 8 g 2 G ;

(2)

or to replace precise invariance with approximative invariance in the same space l11 (G):

9 fn g : n 2 l11 (G) ; kgn ? n k ! 0 8 g 2 G ;

(3)

where k
k is the norm in l1 (G). Condition (2) is the standard de nition of an amenable group , and the equivalent condition (3) is called Reiter's condition [Pa88]. Taking in (3) the measures n = 1A =jAnj; An  G, where 1A is the indicator of a set A, and jAj { its cardinality, gives rise to Flner's condition n

9 fAn g : An  G ; jgAjnA4jAn j ! 0 8 g 2 G ; n

(4)

which is also equivalent to amenability [Pa88], where jAj denotes the cardinality of a set A. The sequences fAng (resp., individual sets An) are called the Flner sequences (resp., the Flner sets ). Given a generating set K  G, it is sucient to check Flner's condition (4) just for all g 2 K, which leads to a characterization of nitely generated amenable groups in terms of isoperimetric properties of their Cayley graphs: nj 9 fAn g : An  G ; j@A jAnj ! 0 ;

(5) where

@A = @K A = fg 2 A : there is a neighbour h 2= Ag = fg 2 A : Kg 6 Ag

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES

3

is the boundary of A in the (left) Cayley graph GK of G determined by a nite generating set K. Property (5) can be formulated for an arbitrary locally nite graph. A graph with uniformly bounded vertex degrees is called amenable if it satis es condition (5), i.e., has subsets with arbitrarily small isoperimetric ratio j@Aj=jAj (sometimes such graphs and these subsets are also called Flner). Amenable graphs are characterized in spectral terms as the graphs for which the spectral radius of the Markov operator P of the simple random walk is 1 [Ge88], or, equivalently, for which 0 belongs to the spectrum of the the discrete Laplacian P ? I (this is no longer true without the assumption of uniform boundedness of vertex degrees [Ka92] which is an analogue of the bounded geometry condition used in Riemannian geometry). The latter property is a discrete counterpart of the condition that 0 belongs to the spectrum of the Laplace{Beltrami operator of a Riemannian manifold (see below Section 8). 2. Discrete equivalence relations

Let (X; ) be a non-atomic Lebesgue measure space (i.e., isomorphic to the unit interval with the Lebesgue measure on it), and R  X  X { an equivalence relation on X. The multiplication (x; y)(y; z) = (x; z) determines a groupoid structure on R. Denote by [x] = [x]R the R-equivalence class (the leaf ) of a point x 2 X. We shall assume that R is countable non-singular , i.e., the classes [x] are at most countable, R is a measurable subset of X  X, and for any subset A  X with S (A) = 0 its saturation [A] = x2A [x] also has measure 0 (the latter means that the measure  is quasi-invariant with respect to R). For simplicity we shall usually assume that R is ergodic , i.e., all measurable R-saturated sets have measure either 0 or 1. Integrating the counting measures on the bers of the left (x; y) 7! x and the right (x; y) 7! y projections from R onto X by the measure  gives the left  y) = dM(y; x) = d(y) counting measures dM(x; y) = d(x) and the right dM(x; on R, respectively. The measures M and M are equivalent i  is quasi-invariant,  y) is called the in which case the Radon{Nikodym derivative D(x; y) = dM=dM(x; Radon{Nikodym cocycle of the measure  with respect to R. Equivalently, the measure  is quasi-invariant i for any partial transformation ' of R (i.e., a measurable bijection between two measurable sets A; B  X whose graph is contained in R) the measure '(jA ) is absolutely continuous with respect to jB , and then d'(jA)=djB (y) = D('?1 (y); y). Thus, D(x; y) is a \regularization" of the formal expression d(x)=d(y); (x; y) 2 R. In particular, if R is the orbit equivalence relation determined by a measure type preserving action of a countable group G on (X; m), then D(x; gx) = dg=d(gx). If D  1, then the measure  is called R-invariant. The equivalence relation (X; ; R) is said to have type I if its classes are a.e. nite, type II1 if it has an equivalent nite invariant measure, type II1 if it has an equivalent - nite invariant measure, and type III otherwise [FM77]. For any measurable set A  R M(A) =

Z

Z

jAx j d(x) = 1A (x; y) dM(x; y) =

Z

Z

1A(x; y)D(x; y) dM(y; x) = jAy jy d(y) ;

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VADIM A. KAIMANOVICH 







where Ax = y : (x; y) 2 A and Ay = x : (x; y) 2 A are the left and the right cross-sections of A, and j
jy is the measure on the class [y] de ned as (6) jxjy = D(x; y) = d(x)=d(y) : In other words, the weights jxjy of the measure j
jy are \proportional" to d(x). By the cocycle property of the Radon{Nikodym derivatives D(
;
) the measures j
jy corresponding to equivalent points y are all pairwise proportional. 3. Amenability of equivalence relations

Type I equivalence relations are characterized by existence of nite leafwise invariant measures (7) 9 fmx gx2X : mx 2 l11 [x] ; mx = my for M-a.e. (x; y) 2 R ; where the map x 7! mx is measurable in the sense that the function (x; y) 7! mx (y) on R is measurable. In complete analogy with the group case one can introduce two generalizations of condition (7): (8) 9 fpxgx2X : px 2 (l1 )1 [x] ; px = py for M-a.e. (x; y) 2 R ; and (9) 9 fnx gx2X;n=1;2;::: : nx 2 l11 [x] ; knx ? ny k ! 0 for M-a.e. (x; y) 2 R ; where the maps x 7! px in (8) and x 7! nx ; n = 1; 2; : : : in (9) are supposed to be measurable: the former in the sense that the function x 7! px (F) is measurable for any F 2 L1 (X; ), and the latter in the same sense as in (7). An equivalence relation (X; ; R) satisfying condition (8) is called amenable (cf. [Zi77], [CFW81]). Although it is quite easy to check that condition (9) is equivalent to (8), this condition (at least in an explicit form) seems to be new. The advantage of condition (9) is in its constructivity, and it signi cantly clari es and simpli es a number of results connected with amenability of equivalence relations and group actions. In particular, (9) can be used to give a new more geometric proof of the theorem of Connes{Feldman{Weiss [CFW81] on equivalence of amenability and hyper niteness (also see Section 7). We shall return to this subject elsewhere. 4. Graphed equivalence relations

A (non-oriented) graph structure on the classes of an equivalence relation (X; ; R) is given by a symmetric measurable subset K  R in such way that two points x; y 2 X are joined with an edge i (x; y) 2 K. We shall call (X; ; R; K) a graphed equivalence relation [Ad90]. Actually, in a somewhat less explicit form (in terms of nitely generated pseudogroups) this de nition is already present in [Pl75], [Se79] and [CG85]. Denote by [x]K the equivalence class [x] endowed with the graph structure K. Then for any x 2 X the cross-section Kx = K x of K is the set of neighbours of x in the graph [x]K . By @A = @K A = fy 2 A : Ky 6 Ag

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES

5

we denote the boundary of a subset A  [x] determined by the graph structure [x]K . The power K n of K with respect to the groupoid operation in R is the set of all pairs of points (x; y) 2 K such that y can be joined with x by precisely n graph edges. The leafwise graph metrics d = dK on classes of R (determined by the graph structure K) are given by the formula d(x; y) = minfn  0 : (x; y) 2 K n g. We say that a graph structure K is connected if it generates the groupoid R, i.e., if the graphs [x]K are a.e. connected. Any set of partial transformations 'i generating the groupoid of R determines a connected graph structure K as a union of graphs of 'i 's and their inverses, and, conversely, any connected graph structure can be presented in this way. If R is the orbit equivalence relation of an action of a countable group G, then any symmetric subset K0  G determines the graph structure (10)



K = (x; gx) : x 2 X; g 2 K0



on R. If the action of G is free, then the maps g 7! g?1 x are isomorphisms between the (right) Cayley graph of G determined by K0 and the graphs [x]K , in particular, the metrics dK coincide with the word metric determined by K0 . Although any equivalence relation is the orbit equivalence relation of a certain group action [FM77] (not necessarily free, as it follows from recent results of Furman [Fu97]), there are graph structures which look quite dierently from Cayley graphs or their quotients, see examples in Section 6 below. We call a graph structure K  R bounded if the graphs [x]K have uniformly bounded vertex degrees (i.e., the cardinalities of the cross-sections of K are uniformly bounded) and the Radon{Nikodym derivatives D(x; y); (x; y) 2 K are uniformly bounded. Note that the question of existence of equivalence relations whose groupoid is not nitely generated seems to be open. Such equivalence relations must be of type II1 (B. Weiss, private communication). 5. Global amenability and local geometry

There are two notions of amenability for an equivalence relation (X; ; R) with a bounded graph structure K. The \global" amenability is given by equivalent conditions (8) and (9) and does not depend on the graph structure K, whereas by the \local" or \leafwise" amenability we mean that -a.e. graph [x]K satis es condition (5), i.e., has a Flner sequence. What are relations between the global and the local amenability? More generally, what are connections between the global amenability and the \local" (leafwise) geometry? One can also formulate these questions in the continuous (rather than measuretheoretical) category for foliations . There is a well known reduction from the continuous to the measure-theoretical category [Pl75] consisting in choosing a family of

ow boxes covering the foliated space and then considering the induced equivalence relation on the union of transversals to these ow boxes. This equivalence relation and the original foliation are amenable or non-amenable with respect to a given quasi-invariant transverse measure type simultaneously. Moreover, the obtained equivalence relation can be given the graph structure generated by the transition

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VADIM A. KAIMANOVICH

maps between the ow boxes. If the foliated space is compact, or, more generally, the foliation has bounded geometry, the resulting leafwise graphs are roughly isometric to the leaves, and therefore have the same isoperimetric properties (see Section 8 for more details). The rst result connecting global amenability with local properties of leaves was that of Series [Se79] and Samuelides [Sa79] who proved that foliations of polynomial growth are amenable (actually, they established hyper niteness of these foliations later shown to be equivalent to amenability by Connes{Feldman{Weiss [CFW81]). It was done by using the above reduction to equivalence relations and proving that graphed equivalence relations of polynomial growth are hyper nite. Brooks in [Br83] formulated the following Example{Theorem 4.3: \Let F be a foliation with invariant measure . If -almost all leaves are Flner, then F is amenable with respect to ". The only indication to the proof is the following phrase: \The proof of this under the assumption that -almost all leaves are of polynomial growth was carried out by C. Series in [Se79]. The argument goes through with some technical changes to yield 4.3". A later paper by Carriere{Ghys [CG85] contains the following Theoreme 4: \Soit (M; F ) un feuilletage d'une variete compacte possedant une mesure transverse invariante  et dont -presque toutes les feuilles sont sans holonomie. Alors F est moyennable pour  si est seulement si -presque toutes ses feuilles sont Flner". In the \Esquisse de demonstration" the authors say that \ : : : la preuve de la condition susante est donnee dans [Br83]" without any further comments. As for the necessary condition, its proof in [CG85] uses the measure-theoretical reduction. Further on, the paper [HK87] by Hurder{Katok deals explicitly with the measure theoretical category. Given a Lebesgue space (X; ), the authors consider what they call a \metric equivalence relation" F on it (in particular, graphed equivalence relations with leafwise graph metrics d belong to this class), and state explicitly the following claim in Proposition 1.3 : \If a Flner sequence exists for almost every x 2 X, then (X; F ; d) is an amenable equivalence relation". The authors use here a strengthened form of the Flner condition: they require the Flner sets to form an increasing sequence exhausting the whole space. Once again, no proof is given, and the reader is referred to the paper [CFW81] by Connes{Feldman{Weiss. In fact, although Lemma 8 from [CFW81] contains a certain Flner type condition equivalent to amenability of an equivalence relation, this condition by no means coincides with the leafwise Flner condition (see below Theorem 2). One of the aims of the present paper is to show that, generally speaking, global amenability can not be deduced from the local amenability (the leafwise Flner condition) in the measure-theoretical category. Namely, in the next Section we give several examples of graphed equivalence relations proving the following Theorem 1. There exists a non-amenable type II1 equivalence relation (X; ; R) with a connected bounded graph structure K  R such that -a.e. graph [x]K ; x 2 X is amenable. In particular, we exhibit a non-amenable graphed equivalence relation whose leaves satisfy the strengthened Flner condition (Example 4), thus disproving Proposition 1.3 from [HK87]. As for the continuous category, we give an example of a

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES

7

type II1 non-amenable foliation of a compact manifoldwith Flner leaves (Theorem 3). Nevertheless, the question about the corresponding II1 example (i.e., about a foliation of a compact manifold with Flner leaves non-amenable with respect to a certain nite transverse invariant measure) apparently remains open (see Section 8 for details). In view of condition (9), there is nothing surprising in Theorem 1. Indeed, the reason for amenability of the orbit equivalence relation of any measure type preserving action of an amenable group is that any Flner sequence on the group immediately determines a sequence of approximatively invariant measures nx in condition (9). On the contrary, in the non-homogeneous case there is no way for leafwise Flner sequences to produce approximatively invariant families of measures nx . The point is that unless R is of type I, any measurable family of leafwise probability measures x has to depend on x in a non-trivial way. However, replacing the leafwise Flner condition with the stronger leafwise subexponential growth condition easily allows one to construct approximatively invariant sequences of measures and to generalize the results from [Sa79] and [Se79]. A graph ? is said to have a subexponential growth if limlog jB(x; n)j=n = 0 for any vertex x 2 ?, where B(x; n) is the n-ball in ? centered at x. Obviously, for any such graph the sequence of balls B(x; n) contains a Flner subsequence. Proposition 1. Let (X; ; R) be an ergodic countable non-singular equivalence relation with a connected bounded graph structure K  R. If a.e. graph [x]K ; x 2 X has subexponential growth, then R is amenable. Proof. For a point x 2 X let xn be the uniform distribution on the ball B(x; n) in [x]K . Then for any (x; y) 2 R with d(x; y) = r

kxn ? ynk  kxn ? xn?r k + kyn ? xn?r k     j B(x; n ? r) j j B(x; n ? r) j = 2 1 ? jB(x; n)j + 2 1 ? jB(y; n)j   j B(x; n ? r) j  4 1 ? jB(x; n + r)j : Therefore, the Cesaro averages nx = (x1 + x2 +


+ xn )=n satisfy condition (9): n X

jB(x; n ? r)j  4 ? 4(n ? r) j n k=r+1 B(x; n + r)j  4 ? 4(n ? r) jB(x; n + r)j ?+2 ! 0

knx ? ny k  4 ? n4

n

n Y

jB(x; n ? r)j j k=r+1 B(x; n + r)j

! n?1 r

r n r

 This result (in a more general context of universal amenability ) is ascribed by Adams{Lyons in [AL91] to an unpublished work of Dougherty and Kechris (1988). Our proof is a \constructivization" of the argument from [AL91], Proposition 3.3.

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VADIM A. KAIMANOVICH

6. Non-amenable equivalence relations with amenable leaves

Example 1. Let (X0 ; 0; R0) be a non-amenable measure preserving equivalence relation with a bounded graph structure K0 . For example, take a free measure preserving action of a nitely generated non-amenable group with leafwise Cayley graph structures (10). Denote by (X; ; R) the suspension over (X0 ; 0; R0) determined by a measurable function ' : X0 ! Z+, i.e., 



X = (x; n) : x 2 X0 ; 0  n  '(x) ; the measure  is de ned as d(x; n) = d0(x), and the classes of R are 







(x; n) = (y; k) 2 X : y 2 [x] :

Then the equivalence R relation R is also non-amenable, the measure  is R-invariant, and it is nite i '(x) d0(x) < 1. We de ne a bounded graph structure K  R as (11)

n?




o n?




o

(x; 0); (y; 0) : (x; y) 2 K0 [ (x; n ? 1); (x; n) : x 2 X0 ; 1  n  '(x) :

Geometrically it means that we add to each vertex x 2 X0 a segment of length '(x) \sticking out" of x. If the function ' is unbounded, then a.e. graph [x]K contains arbitrarily long segments, and is thereby amenable. Example 1 being somewhat \degenerate", it can be easily modi ed. ?
?
 Example 2. Add to the set K (11) all pairs x; '(x) ; y; '(y) with (x; y) 2 K0 . Then the resulting graphs [(x; 0)]K can be visualized as two copies of the graph [x]K0 connected by segments whose length is determined by the values of the function ' on the vertices of [x]K0 . In particular, the graphs [(x; 0)]K do not have pendant vertices. A rooted tree is the couple (T; x), where T is a tree, and the root x is a vertex of T. Denote by T the space of (isomorphism classes of) rooted locally nite trees, and say that two rooted trees are equivalent if they are isomorphic as unrooted trees. Classes of the resulting equivalence relation R are given a natural graph structure, but in general they may contain loops (determined by non-trivial tree automorphisms). Let T0 be the subset of T corresponding to rigid trees (those with a trivial automorphism group). Then the R-equivalence class of any  2 T0 has a tree structure isomorphic to . By a theorem of Adams [Ad90], (T0; R) is non-amenable with respect to any nite invariant measure  concentrated on trees with more than 2 ends. Thus, if  is any such measure, and -a.e. tree contains arbitrarily long segments without branching, then a.e. leaf is amenable, but the equivalence relation is not. We shall now give two examples of such measures. Example 3. Fix a non-degenerate (not concentrated on a single point)Pprobability distribution fpi g on the set f0; 1; 2; :: : g with a nite rst moment 1 < i ipi < 1. The distribution fpig determines a supercritical Galton{Watson branching process and, after conditioning by non-extinction, a random in nite family tree rooted at

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES

9

the progenitor (a Galton{Watson tree ). Denote by  the corresponding Galton{ Watson measure on T0 (the distribution of Galton{Watson trees), and by e the augmented Galton{Watson measure on T0 , which is de ned just like  except that the number of children of the root (only) has the distribution p0i = pi?1 (i.e., the root has i+1 children with probability pi ), and these children all have independent standard Galton{Watson descendant trees with ospring distribution fpi g [LPP95]. e Then -a.e. tree is rigid and has a continuum of ends, and the nite measure e d(x) = d(x)= deg x (where deg x is the degree of the root of a rooted tree x) is R-invariant and ergodic [Ka98]. If p1 > 0, then -a.e. tree has arbitrarily long geodesic segments without branching. The next example is in a sense dual to the previous one. Randomness here is introduced by \stretching" edges of a homogeneous tree (this model was rst considered in [AL91]). Example 4. Fix a non-degenerate probability distribution fpig; i  1. Denote by E the set of (non-oriented) edges of a xed homogeneous rooted tree T of degree 4 which we identify with the Cayley graph of the free group F2 with 2 generators. Consider a family of independent p-distributed random variables fl" g"2E , and denote by P the corresponding probability measure on the space ZE+ of integer-valued con gurations on E (i.e., P is the Bernoulli measure over E determined by the distribution fpig). The measure P is invariant and ergodic with respect to the free action of a non-amenable group F2 on ZE+ by translations, so that the corresponding orbit equivalence relation is non-amenable. Denote by  : ZE+ ! T the map which assigns to a con guration fl" g"2E the tree obtained from T by replacing each edge " with a segment of length l" and rooted at the origin of T. Then the measure 1 = (P) is concentrated on T0. Although the measure 1 itself is not R-quasi-invariant (1 -a.e. tree has vertices of degree 2, whereas the measure 1 of trees rooted at such vertices is 0), it can be easily augmented to an R-invariant measure  which is nite i the distribution fpig has P a nite rst moment ipi [Ka98]. Ergodicity of the measure  with respect to R follows from ergodicity of P. Non-amenability of (T0 ; R; ) (which follows at once from the theorem of Adams) can be also deduced from non-amenability of the action of F2 on (ZE+; P). On the other hand, if the distribution fpig is not nitely supported, then once again -a.e. tree has arbitrarily long geodesic segments without branching. Actually, in this construction one could take an arbitrary measure on ZE+ invariant with respect to the action of the group of automorphisms of T and whose one-dimensional distribution has a nite rst moment. Example 4 also shows that for a.e. graph [x]K the Flner sets An can be chosen increasing and exhausting the graph . The latter condition is sometimes imposed in addition to the standard formulation (5) of the Flner property (e.g., see [HK87]). Denote by Sk the set of vertices of T at distance k from the origin, and by Ek the set of edges joining vertices from Sk and Sk+1 . Let  Zk = (l" )"2E : there exists an edge " 2 Ek with l" > k2 g : Then P(Zk) = 1 ? (p1 + p2 + : : :pk2 )jE j ; k

10

VADIM A. KAIMANOVICH

and one P can easily choose a distribution fpi g with a nite rst moment in such way that k P(Zk ) = 1 (for example, any distribution with a polynomial decay will do). Since the events Zk are independent, by the Borel{Cantelli lemma for P-a.e. con guration fl" g"2E there exist in nitely many indices ki and edges "i 2 Sk with l" > ki2. For a xed con guration fl" g"2E with this property we shall now construct ? in-
ductively an increasing exhausting sequence An of subsets of the tree x =  fl" g with the property (5). We begin with A0 consisting just of the origin of x. Given a subset An let Bn be the minimal ball centered at the root of x and containing An . Then take a suciently large index ki (to be speci ed later), denote by "ei the segment in x obtained by stretching "i , and take An+1 to be the union of Bn and the geodesic joining the root of x with the farthest from the root endpoint of "ei . Then jAn+1j  jBn j +ki2 provided "ei does not intersect Bn , whereas j@An+1j  jBn j +ki, so that we can indeed choose ki in such way that j@An+1j=j@Anj < 1=n. Remark. The construction above is based on the following simple observation. Let ? be a graph with uniformly bounded vertex degrees. Fix a reference point o 2 ?, ?
and let An  ? be a Flner sequence such that j@An j + d(o; An) =jAnj ! 0. Then there exists an increasing Flner sequence exhausting ?. Indeed, add to the sets An geodesic segments joining them with o, then the resulting sequence A0n is still Flner and all sets contain o. Since for any nite set B  ? the sequence A0n [ B is also Flner, we can now proceed inductively and obtain an exhaustive increasing Flner sequence. i

i

7. An isoperimetric criterion of amenability

Examples proving Theorem 1 were based on a principal dierence between global and local amenability. The global amenability of an equivalence relation is inherited when passing to the restriction of the original equivalence relation to a smaller subset (for example, it follows at once from condition (9)). On the other hand, local amenability, i.e., amenability of individual graphs does not have this property: a subgraph of an amenable graph may well be non-amenable. Another important point complicating the relationship between the global and the local amenability is the role of invariance of the measure . Generalizing the fact that any nite measure preserving free action of a non-amenable group is nonamenable, Carriere{Ghys proved that if the measure  is invariant with respect to an amenable equivalence relation R, then for any bounded graph structure K  R a.e. graph [x]K is amenable (the necessity part of Theoreme 4 in [CG85]). This is no longer true if the measure  is not R-invariant. For instance, there are well known examples of amenable orbit equivalence relations arising from free actions of non-amenable groups without invariant measure. The simplest example is the action of a nitely generated free group on the space of ends of its Cayley graph (see below Example 5). The following result shows that these are the only reasons for the discrepancy between the global and the local amenability. Its proof (see also [Ka97]) is based on using condition (9) and a reformulation of the Flner type condition introduced in [CFW81], Lemma 8.

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES

11

Theorem 2. Let (X; ; R) be an ergodic countable non-singular equivalence relation with a connected bounded graph structure K  R. Then R is amenable if and only if for any non-trivial measurable set X and a.e. point x 2 X there exists a family of nite subsets An  [x] \ X such that 0

0

j@K0 An jx ! 0 ; jAnjx where K0 = K \ X0  X0 is the restriction of the graph structure K to X0 , and j
jx is the measure on [x] de ned in (6). We call the sequence of sets An from Theorem 2 Flner with respect to the leafwise measures j
jx . Note that the graph structure K0 on the equivalence relation R0 = R \ X0  X0 is not necessarily connected. For a nite connected component A of a graph [x]K0 the isoperimetric ratio is 0, because the boundary @K0 A is empty. Corollary. If (X; ; R) is an ergodic countable non-singular amenable equivalence relation with a connected bounded graph structure K  R, then a.e. leaf [x]K has a sequence of Flner sets with respect to the measure j
jx . If the measure  is invariant, then all measures j
jx are counting, and we obtain as a particular case the necessity part of Theoreme 4 from [CG85]. Example 5. Denote by @F2 the space of ends (the space of in nite words ) of the free group F2 with two generators. Let  be the equidistributed probability measure on @ F2 , i.e., such that the measures of all cylinders consisting of in nite words with xed rst n letters are equal. Then the orbit equivalence relation R of the free action of F2 on @ F2 is amenable with respect to  (actually, with respect to any purely non-atomic quasi-invariant measure). The simplest explanation is that R coincides with the orbit equivalence relation of the unilateral shift in the space of in nite words [CFW81]. We identify the classes [x] of R with G by the map g 7! g?1 x (provided x has a trivial stabilizer in F2), and endow them with the Cayley graph structure (10). Denote by bx the Busemann function on F2 with respect to the point x 2 @ F2 de ned as h ?
i
? ; bx(g) = lim d g; x [n] ? d e; x[n] n

where d is the Cayley graph distance in F2 , and (e; x[1] ; x[2]; : : :) is the geodesic ray joining the group identity e and the point x, i.e., x[n] consists of n initial letters of the in nite word x. The level sets Hk (x) = fg 2 F2 : bx(g) = kg of the function bx are the horospheres in F2 centered at x. The sign in the de nition of the Busemann function is chosen in such way that the Busemann function goes to ?1 along geodesic rays which converge to x, so that the larger is the index k of the horosphere Hk (x), the farther it is from x. The Radon{Nikodym cocycle of the measure  is D(g?1 x; x) = dg=d(x) = 3?b (g) : x

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Thus, amenability of the equivalence relation R implies that given the measure

jgjx = jg?1xjx = D(g?1 x; x) = dg=d(x) = 3?b (g) x

on F2 (the image of the measure (6) under the map g 7! g?1 x), there exist Flner sequences in the Cayley graph of F2 with respect to this measure (although the Cayley graph of the free group F2 does not have usual Flner sequences). In our case one can easily exhibit these Flner sets explicitly. Namely, let An = fg 2 F2 : 0  bx (g) = d(e; g)  ng be the set of all words g of length  n such that their rst letter does not coincide with x[1] . Then intersections of An with the horospheres Hk (x); 0  k  n all have the same measure j
jx equal 1, so that jAnjx = n + 1. On the other hand, 



@An = feg [ An \ Hn(x) ; and j@Anjx = 2. 8. Non-amenable foliations with amenable leaves

Let F be a codimension k foliation of a compact manifold M of dimension n. Fix a family of ow boxes i : Ui ! D k  D n?k (here D d is the d-dimensional open disk) indexed by a nite set I and covering M. We may assume ? that this
family is regular in the sense that for any given i 6= j 2 I any plaque i?1 fz gD n?k in ?Ui intersects
at most one plaque in Uj . Fix on each Ui the transversal Ti = ?1 Dk  f0g , and let T be the disjoint union of Ti ; i 2 I. Then we obtain a family of partial dieomorphisms (transition maps) ij de ned as ij x = y if x 2 Ti ; y 2 Tj and the plaques through i and j meet. The pseudogroup of partial dieomorphisms of T generated by f ij gi;j 2I is contained in the full holonomy pseudogroup of F and is called the fundamental pseudogroup of F determined by the ow boxes Ui [Pl75], [Br84]. Denote by R = R(F ; T) the equivalence relation on T obtained by restricting the foliation equivalence relation to T: two points x; y 2 T are equivalent i they belong to the same leaf of the foliation. Then ij are partial transformations which generate the equivalence relation R. Denote by K the corresponding graph structure on R which is the union of graphs of ij . The graphed equivalence relation (T; R; K) has the same structure properties as the original foliation F . In particular, the restriction  = jT of any holonomy quasi-invariant (resp., invariant) measure  to T is quasi-invariant (resp., invariant) with respect to R, and, conversely, any quasi-invariant (resp., invariant) measure of R extends to a holonomy quasi-invariant (resp., invariant) measure of F . The foliation F is called amenable with respect to a holonomy quasi-invariant measure  if R is amenable with respect to the corresponding measure . This property does not depend on the choice of the ow boxes Ui . The leaf Lx passing through a point x 2 T is called Flner (or, amenable in our terminology) if the corresponding graph [x]K is amenable in the sense of the

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de nition from Section 1. If the leaves of F are endowed with the Riemannian metrics induced by a Riemannian metric on the foliated manifold M, then a leaf Lx is amenable i 0 belongs to the spectrum of the leafwise Laplacian on Lx , or, equivalently, i the Cheeger isoperimetric constant of Lx is 0 [Br84]. Any two global Riemannian metrics on a compact manifold are quasi-isometric, therefore leafwise amenability does not depend on the choice of a particular Riemannian metric on M. Actually, more general isoperimetric properties are also invariant with respect to rough isometries (see [Kn85], so that they are the same for leafwise Riemannian metrics and for the corresponding leafwise graphs. Note that if Lx is an amenable leaf, then any weak limit point of the sequence of measures n = 1A =jAnj (where fAng is a Flner sequence in the graph [x]K ) is an R-invariant measure [GP79]. If M is non-compact one can still perform this reduction from continuous to the measure-theoretical category provided the leaves of F are given Riemannian metrics of uniformly bounded geometry. In the same way as before one can ask in this topological setup about the relationship between the global and the local amenability of a foliation F with respect to a quasi-invariant measure . As we have just seen, this question is equivalent to the same question about the induced equivalence relation R on the transversal T with respect to the measure  = jT . It is this reduction to the measuretheoretical category which was referred to in [Br83], Example-Theorem 4.3 (see also [CG85], Theoreme 4) when claiming that if the measure  is invariant then the local amenability implies the global amenability. Thus, Theorem 1 shows that this argument is incomplete, and the following question remains open. Problem 1. Let F be a foliation of a compact manifold with a nite transverse invariant measure  such that -a.e. leaf is Flner. Is F amenable with respect to ? S. Hurder suggested a variant of this problem where the measure  is supposed to be obtained from a certain leafwise Flner sequence. One might also require minimality of F . Problem 1 is closely related to the following more general question formulated in the author's paper [Ka88]: Problem 2. Let F be a foliation of a compact manifold with a harmonic measure  such that -a.e. leaf is Flner. Does -a.e. leaf have the Liouville property ( no non-constant bounded harmonic functions)? Problems 1 and 2 can be also formulated in a more general setup of the equivan

lence relations generated by a nite number of partial homeomorphisms of a compact set with a nite invariant measure . A positive answer to Problem 2 would imply a

positive answer to Problem 1 (for, any foliation with Liouville leaves is amenable, e.g., see [CFW81], Proposition 20). It would also imply that for a foliation of compact manifold with Flner leaves any harmonic measure is completely invariant (i.e., corresponds to a transverse invariant measure), see [Ka88], Corollary of Theorem 4. For foliations with leaves of subexponential growth the answer to Problem 2 is positive [Ka88], Theorem 2. Therefore, such foliations are amenable with respect to all harmonic measures, and, moreover, any harmonic measure is completely invariant. Note that amenability of foliations with leaves of subexponential growth with

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respect to any transverse quasi-invariant measure follows from the direct argument given in Proposition 1. Of course, the examples from Section 6 can be easily recast to provide type II1 Riemannian measurable foliations (in the sense of [Zi83], i.e., those with a Riemannian leafwise and a measurable transverse structures) giving a negative answer to Problems 1 and 2 (also see [Ka88]). We shall now give an example (inspired by several discussions with S. Hurder) of a type II1 foliation disproving Proposition 1.3 from [HK87] in the continuous category (the question about type II1 -foliations is formulated in Problem 1 above). Theorem 3. There exists a C 1 codimension 2 dimension 2 foliation F of a compact manifold with a - nite invariant measure  such that F is non-amenable with respect to , but -a.e. leaf is amenable. Proof. The example proving Theorem 3 will be constructed by taking a connected sum of two foliations F1 and F2 such that F1 is non-amenable and F2 has amenable leaves roughly in the same as in Example 1 (although here we attach semi-in nite rather than nite segments). The only diculty is to choose F2 in such way that amenability of its leaves is not lost when passing to the connected sum. We begin by choosing a compact manifold M whose fundamental group 1(M) acts on the 2-dimensional torus T2 preserving the Lebesgue measure m. One could use here the fact that any nitely presented group is the fundamental group of a certain compact 4-dimensional manifold, and take for M a 4-manifold with the fundamental group SL(2; Z). However, in order to reduce the dimension of our example, we take a genus 2 compact surface M (I owe this suggestion to F. Alcalde-Cuesta). Then its fundamental group 1(M) is determined by 4 generators a1; a2; b1; b2 and the relation [a1; a2][b1; b2] = e. Let H be the normal subgroup of 1(M) determined by the relations a2 = b2 = e. Then the quotient G = 1 (M)=H is a 2-generator free group. Realizing G as a nite index subgroup of SL(2; Z), we obtain a homomorphism ' : 1(M) ! SL(2; Z), which in combination with the standard action of SL(2; Z) on T2 determines the sought for action  of 1(M) on T2. f the universal covering manifold of M and consider the product Denote by M 2 f as a foliation F e1 with the leaves fz g  M; f z 2 T2. Taking the quotient of T M f with respect to the diagonal action of 1(M), we obtain the foliation F1 . T2  M In other words, F1 is the suspension of the action  over M [CN85]. The leaves of f by the action F1 are dieomorphic to the G-cover of M, i.e., to the quotient of M of the normal subgroup H (except for a countable number of leaves corresponding to the points z 2 T2 with non-trivial stabilizers in G). Denote by T1 the transversal of F1 obtained by taking the image of the transverf Then T1 is sal Te1 = T2  fxg of Fe1 (where x is a chosen reference point in M). 2 naturally dieomorphic to T , and there is a small tubular neighbourhood O(T1 ) of T1 dieomorphic to T2  D 2 by a dieomorphism preserving the foliation structure. The equivalence relation R(F1 ; T1) induced on T1 coincides with the orbit equivalence relation of the action , i.e., with the orbit equivalence relation of the free action of the non-amenable group '(1(M)), which is non-amenable with respect to the invariant measure m. Therefore, F1 is non-amenable with respect to the invariant measure 1 determined by m.

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Recall that the Reeb foliation FR is a codimension 1 C 1-foliation of the sphere S which contains a unique compact leaf T2, and all other leaves are dieomorphic to R2 (e.g., see [Ta92]). The Reeb foliation has a transversal circle TR which does not meet the compact leaf and meets all other leaves at most once. Take for F2 the product of FR and the circle T1. Then T2 = TR  T1 is a transversal of F2 dieomorphic to T2, and such that there is a small tubular neighbourhood O(T2 ) dieomorphic to T2  D 2 by a dieomorphism preserving the foliation. Now we are able to glue together the complements of the neighbourhoods O(T1 ) and O(T2 ) via a dieomorphism which preserves the foliation, which gives a new foliation F = F1 ? F2 (a connected sum of F1 and F2 ). Let T be a transversal of F obtained from this procedure. There are 3 kinds of leaves in F : the compact leaf T2 (coming from the compact leaf of FR ), the leaves dieomorphic to R2 (coming from those non-compact leaves of the Reeb foliation which do not intersect TR ), and the \glued leaves". The latter are the leaves of F1 with \holes" cut around the intersections with the transversal T1 , to which are attached semi-in nite \cylinders" (leaves of F2), so that these leaves are amenable. Moreover, these leaves admit an increasing exhausting sequence of Flner sets obtained by taking unions of balls with long segments of the cylinders. Note that the measures obtained from leafwise Flner sequences are concentrated on the compact leaf. Any regular family of

ow boxes of F contains ow boxes intersecting the compact leaf, so that the associated leafwise graphs are easily seen to contain components (corresponding to the attached cylinders) roughly isometric to Z+. On the other hand, since the leaves of F2 intersect T2 at most once, the equivalence relations R(F ; T) and R(F1; T1 ) coincide (under the natural identi cation of T1 and T), so that F is non-amenable with respect to the invariant measure  induced by the Lebesgue measure m on T2. Note that the measure  is not nite on any transversal intersecting the compact leaf because of the \dissipativity" of the Reeb foliation (more precisely, of the holonomy group of the compact leaf), as a result of which each glued leaf meets such a transversal in nitely many times.  Remark. Obviously dimension 2 can not be lowered in the example from Theorem 3 (all dimension 1 foliations are amenable). For codimension 1 foliations nite invariant measures are supported by leaves of polynomial growth [Pl75], and therefore such foliations are amenable with respect to any nite invariant measure [Sa79], [Se79]. However, if the transverse measure is not required to be invariant, then any non-amenable action of a free group by dieomorphisms of the circle leads to an analogous codimension 1 type III example. We do not know whether there exists a codimension 1 type II1 example. Concluding remark. After the present paper had been circulated as a preprint, F. Alcalde-Cuesta constructed an example of a foliation of a compact manifold with a nite transverse invariant measure such that all leaves are Flner but the foliation is not amenable with respect to this invariant measure. The transverse invariant measure in this example is singular. Later, E. Ghys gave another example where the transverse invariant measure is actually a transverse volume. These examples completely disprove the claims made by Brooks [Br83] and Carriere{Ghys [CG85] and answer the questions formulated in Problems 1 and 2 above in the negative. The main idea of these examples is the same as in the present paper: one performs 3

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a surgery on a non-amenable foliation with non-amenable leaves in such a way that the leaves become Flner, whereas non-amenability of the foliation is preserved. References [Ad90] [AL91] [Br83] [Br84] [CFW81] [CG85] [CN85] [FM77] [Fu97] [CG85] [Ge88] [GP79] [HK87] [Ka88] [Ka92] [Ka97] [Ka98] [Kn85] [LPP95] [Pa88] [Pl75] [Sa79] [Se79] [Ta92] [Zi77] [Zi83]

S. Adams, Trees and amenable equivalence relations, Ergodic Theory Dynam. Systems 10 (1990), 1{14. S. Adams, R. Lyons, Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations, Israel J. Math. 75 (1991), 341{370. R. Brooks, Some Riemanniann and dynamical invariants of foliations, Progr. Math. (Dierential Geometry, College Park, Md., 1981/1982), vol. 32, Birkhauser, Boston, Mass., 1983, pp. 56{72. R. Brooks, The spectral geometry of foliations, Amer. J. Math. 106 (1984), 1001{1012. A. Connes, J. Feldman, B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), 431{450. Y. Carriere, E. Ghys, Relations d'equivalence moyennables sur les groupes de Lie, C. R. Acad. Sci. Paris Ser. I Math. 300 (1985), 677{680. C. Camacho, A. L. Neto, Geometric Theory of Foliations, Birkhauser, Boston, 1985. J. Feldman, C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289{324. A. Furman, Orbit equivalence rigidity, preprint (1997). E. Ghys, Y. Carriere, Relations d'equivalence moyennables sur les groupes de Lie, C. R. Acad. Sci. Paris Ser. I Math. 300 (1985), 677{680. P. Gerl, Random walks on graphs with a strong isoperimetric inequality, J. Theoret. Probab. 1 (1988), 171{188. S. E. Goodman, J. F. Plante, Holonomy and averaging in foliated sets, J. Di. Geom. 14 (1979), 401{407. S. Hurder, A. Katok, Ergodic theory and Weil measures for foliations, Ann. of Math. 126 (1987), 221{275. V. A. Kaimanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funct. Anal. Appl. 22 (1988), 326{328. V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal. 1 (1992), 61{82. V. A. Kaimanovich, Amenability, hyper niteness and isoperimetric inequalities, C. R. Ac. Sci. Paris, Ser. I 325 (1997), 999{1004. V. A. Kaimanovich, Hausdor dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), 631{660. M. Kanai, Rough isometries and combinatorial approximation of geometries on noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), 391{413. R. Lyons, R. Pemantle, Y. Peres, Ergodic theory on Galton{Watson trees: speed of random walk and dimension of harmonic measure, Ergodic Theory Dynam. Systems 15 (1995), 593{619. A. T. Paterson, Amenabilty, Amer. Math. Soc., Providence, R. I., 1988. J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. 102 (1975), 327{361. M. Samuelides, Tout feuilletage a croissance polynomiale est hyper ni, J. Funct. Anal. 34 (1979), 363{369. C. Series, Foliations of polynomial growth are hyper nite, Israel J. Math. 34 (1979), 245{258. I. Tamura, Topology of foliations: an introduction, American Mathematical Society, Providence, RI, 1992. R. J. Zimmer, Amenable ergodic actions, hyper nite factors, and Poincare ows, Bull. Amer. Math. Soc. 83 (1977), 1078{1080. R. J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math. 105 (1983), 1011{1022.

CNRS UMR 6625, IRMAR, Campus Beaulieu, Rennes 35042, France

EQUIVALENCE RELATIONS WITH AMENABLE LEAVES E-mail address : [email protected]

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