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 hyperniteness,
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 sufcient 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 Classication. 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
2
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 denition 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 denition 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 satises 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) ;
4
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] dened as
(6)
jxjy = D(x; y) = d(x)=d(y) :
In other words, the weights jxjy of the measure jjy 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 signicantly claries and simplies
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
hyperniteness (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 denition 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 satises
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
6
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 hyperniteness 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 hypernite.
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.
8
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 dened 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 dene 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 modied.
?
?
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 innite 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 dened 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
congurations 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 conguration 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.
conguration fl" g"2E there exist innitely many indices ki and edges "i 2 Sk with
l" > ki2.
For a xed conguration 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 specied 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] dened 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 innite 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 innite
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 innite 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
dened 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 innite 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 denition 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
12
VADIM A. KAIMANOVICH
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 dened 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 fij 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
EQUIVALENCE RELATIONS WITH AMENABLE LEAVES
13
denition 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
14
VADIM A. KAIMANOVICH
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-innite
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.
EQUIVALENCE RELATIONS WITH AMENABLE LEAVES
15
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-innite \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 identication
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 innitely 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
16
VADIM A. KAIMANOVICH
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.
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CNRS UMR 6625, IRMAR, Campus Beaulieu, Rennes 35042, France
EQUIVALENCE RELATIONS WITH AMENABLE LEAVES
E-mail address : [email protected]
17