aa r X i v : . [ m a t h . L O ] S e p A BRIEF INTRODUCTION TO AMENABLEEQUIVALENCE RELATIONS
JUSTIN TATCH MOORE Introduction
The notion of an amenable equivalence relation was introduced byZimmer in the course of his analysis of orbit equivalence relations inergodic theory (see [12]). More recently it played an important role inMonod’s striking family of examples of nonamenable groups which donot contain nonabelian free subgroups. If A is a subring of R , define H ( A ) to be the group of all piecewise PSL ( A ) homeomorphisms of thereal projective line which fix the point at infinity. Theorem 1.1. [17] If A is any dense subring of R , then H ( A ) is non-amenable. Moreover, if f, g ∈ H ( R ) , then either h f, g i is metabelianor else contains an infinite rank free abelian subgroup. In particular, H ( R ) does not contain a nonabelian free subgroup. Subsequently, Lodha and the author constructed a finitely presentednonamenable subgroup of H ( Z [1 / √ F ∞ [15].At least from a group-theoretic perspective, the most novel aspect of[17] was the use of Zimmer’s notion of an amenable equivalence relationin the proof of the nonamenability of the groups H ( A ). The purposeof this article is to give a brief survey of the theory of amenable andhyperfinite equivalence relations and illustrate how it can be used toshow that certain discrete groups are nonamenable.The subject matter falls within the broader scope of what is some-times called measurable group theory — the study of groups throughthe analysis of their action on measure spaces. This is in contrastwith geometric group theory , where the emphasis is on actions whichpreserve an underlying geometry. Measurable group theory is closelyaligned with the ergodic theory and dynamics of discrete groups, prob-ability, and descriptive set theory. Further reading can be found in [4], Mathematics Subject Classification.
Primary: 43A07; Secondary: 20F65.
Key words and phrases. amenable, equivalence relation, free group, hyperfinite.This research was supported in part by NSF grant DMS–1262019. I would liketo thank Matt Brin and Clinton Conley for their help while preparing this article. [7], [10], [12] and their references. Further background on descriptiveset theory can be found in [11].This article is organized as follows. After reviewing some backgroundmaterial and fixing some terminology, we will present the definitions ofamenable and hyperfinite equivalence relations in Section 3. This sec-tion will culminate with a theorem connecting these two apparently dif-ferent notions. Section 4 will present several examples of nonamenableequivalence relations. Section 5 will discuss the analogs of the closureproperties of amenable groups in the setting of equivalence relations.These played an important role in the isolation of the group in [14].This article does not contain any new results, although Theorem 4.8below is cast in a more abstract way than in [9]. (It is also to theauthor’s knowledge, the first account of this proof in English.) Thearticle’s goal is to encourage the reader to pursue further reading in,e.g., [12] which contains a much more complete treatment of the subjectmatter presented here. 2.
Preliminaries
Before proceeding, we will fix some terminology. In a few placeswe will refer to the continuity of extended real valued functions. Inthis context, the neighborhoods of infinity are the co-bounded sets.Recall that a
Polish space is a topological space which is separable andcompletely metrizable. The σ -algebra of Borel sets in a Polish spaceis said to be a standard Borel space . A function f between standardBorel spaces X and Y is Borel if preimages of Borel sets are Borel.This is equivalent to the graph of f being a Borel subset of X × Y .It is well known that any two uncountable standard Borel spaces areisomorphic in the sense that there is a bijection f between them suchthat f and f − are Borel.A Borel measure on a standard Borel space is a countably additive σ -finite measure defined on its Borel sets. Such a measure extendsuniquely to the σ -algebra generated by the Borel sets and the subsetsof measure 0 Borel sets. We will generally not distinguish betweenthese measures but note here that measurable will always refer to thelarger σ -algebra.In this article we will write ( X, µ ) is a measured Polish space tomean that X is a Polish space and that µ is a Borel measure on X . Ifin addition the topology on X is generated by the open sets of finitemeasure, then we say that ( X, µ ) is locally finite . If Γ is a topologicalgroup, then we will say that Γ acts continuously on a measured Polishspace (
X, µ ) if:
MENABLE EQUIVALENCE RELATIONS 3 • the map ( g, x ) g · x is continuous and • the maps g µ ( g · E ) are continuous for each measurable E ⊆ X .Notice that this is stronger than the assertion that Γ acts continuouslyon the metric space X .We note some useful facts about measured Polish spaces. Fact 2.1. If ( X, µ ) is a measured Polish space and E ⊆ X is measur-able, then µ ( E ) is the supremum of all µ ( F ) where F is a closed subsetof E . Fact 2.2.
Suppose that ( X, µ ) is a locally finite measured Polish space.If E ⊆ X is measurable and has positive measure, then for every ǫ > there is an open set U ⊆ X such that < µ ( U ) < ∞ and µ ( E ∩ U ) > (1 − ǫ ) µ ( U ) . We will also need the following proposition.
Proposition 2.3.
Suppose that ( X, µ ) is a locally finite measured Pol-ish space and Γ is a metrizable group acting continuously on ( X, µ ) . If E ⊆ X is a measurable set of positive measure, then there is an openneighborhood V of the identity of Γ and an ǫ > such that if g is in V , then µ (( g · E ) ∩ E ) > ǫ .Proof. Let E ⊆ X be given and let U ⊆ X be an open set with0 < µ ( U ) < µ ( E ∩ U ) < µ ( U ) < ∞ . Set ǫ = µ ( U ). Observe by our continuity assumption on the action,we have that for every x in U there is an open W x ⊆ U containing x and a δ x > g to the identity is lessthan δ x , then g · W x ⊆ U . Find a δ > µ ( { x ∈ U : δ x ≥ δ } ) > µ ( U )and define W = S { W x : δ x ≥ δ } , observing that µ ( W ) > µ ( U ).In particular, µ ( E ∩ W ) > µ ( U ). By our assumption that Γ actscontinuously on ( X, µ ), there is an open set V containing the identitysuch that every element of V has distance less than δ to the identity and µ ( g · ( E ∩ W )) > µ ( U ) whenever g is in V . Since µ ( E ∩ U ) > µ ( U )and since g · ( E ∩ W ) ⊆ ( g · E ) ∩ U , it follows that µ ( E ∩ ( g · E ) ∩ U ) > µ ( U ) = ǫ . (cid:3) If X is a standard Borel space, an equivalence relation E on X isBorel if it is Borel as a subset of X . A Borel equivalence relation is saidto be countable if every equivalence class is countable. Notice that while JUSTIN TATCH MOORE this meaning conflicts with the literal interpretation of “countable,”there is never a cause for confusion since for an equivalence relation tobe countable as a set, it must have a countable underlying set and inthis context one is generally only interested in uncountable standardBorel spaces (moreover the two notions coincide if the underlying spaceis countable).The principal example of a countable Borel equivalence relation isas follows: if G is a countable discrete group acting by Borel auto-morphisms on a standard Borel space X , then the orbit equivalencerelation E GX is a countable Borel equivalence relation. That is, ( x, y ) isin E GX if and only if there is a g in G such that g · x = y . In fact allcountable Borel equivalence relations arise in this way. Theorem 2.4. [6] If E is a countable Borel equivalence relation on astandard Borel space, then there is a countable group G and a Borelaction of G on X such that E = E GX . The advantage of working with equivalence relations is in part thatthe notion of a countable Borel equivalence relation is much more flex-ible than that of a group. For instance while subgroups give rise tosubequivalence relations, the converse is not generally true. A moresophisticated example of this is Theorem 1.1: orbit equivalence rela-tions are used, in a sense, to transfer the nonamenability of PSL ( A )to the group H ( A ) even though these groups are quite unrelated froman algebraic perspective.3. Amenable and hyperfinite equivalence relations
Any action of a countable group on a standard Borel space gives riseto a countable Borel equivalence relation and, conversely, any countableBorel equivalence relation can be generated as the orbit equivalence re-lation of some group action. The fundamental problem of this subjectis to understand the extent to which properties of the group which gen-erated a countable Borel equivalence relation are reflected in propertiesof the equivalence relation and vice versa.Our focus in this article will be to develop the properties of equiv-alence relations which are analogs of the group-theoretic property of amenability . Roughly speaking, the notion of an amenable equivalencerelation has the property that every action of an amenable group givesrise to an amenable equivalence relation and a group is amenable onlywhen every orbit equivalence relation is amenable.Now to be more precise. Suppose that (
X, µ ) is Borel measure on astandard Borel space and E is a countable Borel equivalence relation on MENABLE EQUIVALENCE RELATIONS 5 X . We say that E is µ -amenable if there is a µ -measurable assignment x ν x such that: • each ν x is a finitely additive probability measure on X satisfyingthat ν x ([ x ] E ) = 1. • if ( x, y ) ∈ E , then ν x = ν y .By measurable we mean that if A is any measurable subset of X × X ,then x ν x ( { y ∈ X : ( x, y ) ∈ A } )is µ -measurable. While we will generally quantify amenable with ameasure, a Borel equivalence relation is said to be amenable if it is µ -amenable with respect to every Borel measure on the underlyingstandard Borel space.While it is not apparent from the definition, it is true that everyorbit equivalence relation of a countable amenable group acting onstandard Borel space is necessarily µ -amenable with respect to anyBorel measure µ (it is interesting to note that this does not require anyinvariance properties of µ with respect to the group action). This willfollow from Theorem 3.5 below.Next we turn to a seemingly unrelated notion. A countable Borelequivalence relation E on a standard Borel space X is hyperfinite if E is an increasing union of Borel equivalence relations with finite equiv-alence classes. A good example to keep in mind is that of eventualequality on infinite binary sequences: define x = ∗ y if x ( k ) = y ( k ) forall but finitely many k . Notice that this is the union of the equivalencerelations = n defined by x = n y if x ( k ) = y ( k ) for all k ≥ n .The following theorem gives a powerful criterion for verifying hyper-finiteness. Theorem 3.1. [4]
Suppose that X is a standard Borel space and f : X → X is a Borel function which such that f − ( x ) is at most countablefor each x . The smallest equivalence relation E satisfying that, for each x ∈ X , ( x, f ( x )) ∈ E is hyperfinite. Example 3.2. [4]
Define an equivalence relation E all infinite binarysequences by xEy if for some m and n , x ( m + i ) = y ( n + i ) for all i > .This equivalence relation is called tail equivalence and is generated bythe shift map f : 2 ω → ω given by f ( x )( i ) = x ( i + 1) . Example 3.3. [2]
Recall that the real projective line P ( R ) is the col-lection of all lines in R passing through the origin. Identify an elementof P ( R ) with the x -coordinate of its intersection with the line y = 1 ,adopting the convention that the line y = 0 becomes identified with ∞ .This identification gives P ( R ) a natural compact metric topology — it JUSTIN TATCH MOORE is in fact homeomorphic to a circle. An element of
PSL ( R ) can thenbe regarded as a fractional linear transformation t at + bct + d . Define amap Φ : 2 ω → P ( R ) by Φ( x ) = φ ( x ) Φ( x ) = − φ ( ∼ x ) where φ ( x ) = 1 + φ ( x ) φ ( x ) = 11 + φ ( x ) (here ∼ x denotes the bitwise complement of x ). This map preserves thecyclic order and is a quotient map from ω onto P ( R ) . The action of PSL ( Z ) on P ( R ) naturally lifts to an action on ω . The correspond-ing orbit equivalence relation on ω is tail equivalence. In particular,this orbit equivalence relation of PSL ( Z ) ’s action on P ( R ) is hyper-finite. The following gives an important characterization of the hyperfiniteequivalence relations.
Theorem 3.4. [19] [20]
Every Borel action of Z on a standard Borelspace generates a hyperfinite orbit equivalence relation. Conversely,every hyperfinite Borel equivalence relation is the orbit equivalence re-lation of a Borel action of Z . While it is not obvious, it turns out that every hyperfinite Borelequivalence relation is in fact µ -amenable with respect to any Borelmeasure on the underlying space. In fact, a natural weakening capturesthe notation of µ -amenability exactly. If µ is a Borel measure on X ,then we say that E is µ -hyperfinite if there is a µ -measure 0 set Y ⊆ X such that the restriction of E to X \ Y is hyperfinite. Theorem 3.5. (see [12] ) Suppose that X is a standard Borel space, E is a countable Borel equivalence relation on X , and µ is a Borelmeasure on X . The following are equivalent: (1) E is µ -amenable; (2) E is µ -hyperfinite; (3) E = E GX for some µ -measurable action of an amenable group G on X ; (4) E = E Z X for some µ -measurable action of Z on X . This is an amalgamation of several results stated in modern language.It is worth noting that the extent to which the measure 0 sets can beomitted in the previous theorem is a major open problem in descriptiveset theory.
MENABLE EQUIVALENCE RELATIONS 7
Problem 3.6. [4] If E n ( n < ∞ ) is an increasing sequence of hyperfi-nite Borel equivalence relations on a standard Borel space, is S ∞ n =0 E n hyperfinite? Problem 3.7. [20]
Is every orbit equivalence relation of a Borel ac-tion of a countable amenable group acting on a standard Borel spacehyperfinite?
In fact it was only recently that a positive solution to Problem 3.7was proved for the class of abelian groups [8]; the strongest result atthe time of this writing is [18]. While not directly related, Marks hasrecently demonstrated differences between the so-called
Borel context and measure-theoretic context [16].4.
Examples
In this section we will consider a number of examples. Perhaps theeasiest way to generate nonamenable equivalence relations is throughactions of groups which preserve a probability measure.
Theorem 4.1. [10]
Suppose that G is a countable group, ( X, µ ) is astandard Borel space equipped with a Borel probability measure, and E is the orbit equivalence relation of a measure preserving action of G which is free µ -a.e.. The equivalence relation E is µ -amenable if andonly if G is amenable. The following are two typical — but quite different — examples ofsuch actions.
Example 4.2. If G is any countable group and ( X, µ ) is any standardprobability space, then G acts by shift on X G as follows: ( g · x )( h ) = x ( g − h ) . This action preserves the product measure and, unless µ is apoint-mass, is free almost everywhere with respect to the product mea-sure. Example 4.3.
The action of SL ( Z ) on the torus T equipped withLebesgue measure is measure preserving and free λ -a.e.. Since SL ( Z ) contains the free group on two generators, this orbit equivalence relationis not λ -amenable. It is interesting to contrast this previous example with that of thegroup PSL ( Z ) acting on P ( R ) (Example 3.3), which is homeomorphicto the circle. It is well known that PSL ( Z ) contains a free subgroup(even one of finite index) and hence is nonamenable. On the otherhand, we have seen above that the orbit equivalence relation inducedon P ( R ) is just tail equivalence on 2 ω in disguise; in particular it JUSTIN TATCH MOORE is hyperfinite and hence amenable. Notice that, unlike the action ofSL ( Z ) on the torus, there is no standard probability measure on P ( R )which is preserved by the action of PSL ( Z ).It turns out, however, that dense subgroups of PSL ( R ) do producea nonamenable orbit equivalence relation when they act on P ( R ). Theorem 4.4. [9]
Every nondiscrete subgroup of
PSL ( R ) is eithersolvable or else contains a nondiscrete free subgroup on two generators. Theorem 4.5. [9] If Γ is a rank 2 free subgroup of a finite dimensionalLie group G and Γ is nondiscrete, then the orbit equivalence relationof Γ ’s action on G is nonamenable with respect to the Haar measureon G . In the case of Γ = PSL ( Z [1 / Example 4.6.
The matrices α = (cid:18) / − / (cid:19) β = (cid:18) / − /
44 0 (cid:19) generate a nondiscrete free subgroup of
PSL ( Z [1 / . In order to seethis, first observe that the traces of these matrices are / and henceboth matrices describe elliptic transformations of the real projective line(i.e. there are no fixed points). Since any elliptic element of PSL ( R ) of infinite order generates a nondiscrete subgroup, it suffices to showthat the above matrices generate a free group.Define X to be the set of all rational numbers in P ( R ) = R ∪ {∞} which can be represented by a fraction with an odd denominator and let Y denote the remaining rational numbers in P ( R ) . By the Ping-PongLemma (see, e.g., [3] ), it suffices to show that if n = 0 is an integer,then α n Y ⊆ X and β n X ⊆ Y . Let X consist of those elements of X which can be represented by a fraction of the form (4 p + 2) /q where q is odd. Notice that α ( X ∪ Y ) ⊆ X and that X and Y are disjoint.It follows that α n Y ⊆ X whenever n is a nonzero integer. Similarly, β n X ⊆ Y . Theorem 4.4 was generalized considerably by the following result.
Theorem 4.7. [1] If Γ is a dense subgroup of a connected semi-simplereal Lie group, then Γ contains a dense free subgroup of rank 2. The following theorem is a generalization of Theorem 4.5, althoughthe argument closely follows that of [9].
Theorem 4.8.
Suppose that ( X, µ ) is a locally finite measured Pol-ish space. If Γ = h a, b i is a free nondiscrete metrizable group which MENABLE EQUIVALENCE RELATIONS 9 is acting freely and continuously on ( X, µ ) , then the orbit equivalencerelation is not µ -amenable.Proof. Suppose for contradiction that E Γ X is µ -amenable and fix a µ -measurable assignment x ν x such that: • for each x , ν x is a finitely additive probability measure sup-ported on the orbit of x ; • if x and y are in the same orbit, then ν x = ν y .For u ∈ { a, b } , define Γ u to be all those elements of Γ which arerepresentable by a reduced word beginning with u and ending with u − . Observe that if g is a nonidentity element of Γ, then thereis a h in { a, ab, ab − } such that hgh − is in Γ a . Thus there is an h ∈ { e, a, ab, ab − } and a Γ ′ ⊆ Γ which accumulates to the identitysuch that h Γ ′ h − ⊆ Γ a . Since conjugation is continuous, it follows thatΓ a also accumulates to the identity. Furthermore, b Γ a b − ⊆ Γ b andthus Γ b accumulates to the identity as well.Since the action of Γ on X is free, for each x, y ∈ X which lie inthe same orbit, there is a unique γ = γ ( x, y ) in Γ such that x = γ · y .Notice that γ ( g · x, y ) = g − γ ( x, y ). Define φ : X → [0 ,
1] by letting φ ( x ) = ν x ( A x ) where A x is the set of those y in the orbit of x such thatthe reduced word representing γ ( x, y ) begins with a or a − . Observethat for any x in X and g in Γ b , A x ∩ A g · x = ∅ . Hence if x is in X and φ ( x ) > /
2, then φ ( g · x ) < / g is in Γ b . Similarly, if φ ( x ) < / g is in Γ a , then φ ( g · x ) > /
2. Furthermore, for all x and g ∈ Γ a , the sets A x , A bgb − · x and A b gb − · x are pairwise disjoint andconsequently 0 ≤ φ ( x ) + φ ( bgb − · x ) + φ ( b gb − · x ) ≤ Y = { x ∈ X : φ ( x ) = 1 / } has positive measurewith respect to µ . Suppose not. Using our assumption that Γ actscontinuously on ( X, µ ), find an open neighborhood V of the identitysuch that if g is in V , then Y , bg − b − · Y , and b g − b − · Y havetotal measure less than that of X . Now if x is outside these sets and g ∈ V ∩ Γ a , we have that φ ( x ), φ ( bgb − · x ), and φ ( b gb − · x ) are each1 /
2, contradicting that there sum is at most 1. It must therefore bethat Y has positive measure.Let Y a = { y ∈ Y : φ ( y ) > / } and Y b = { y ∈ Y : φ ( y ) < / } .Since Y has positive measure, either Y a or Y b have positive measure. If Y a has positive measure, then by Proposition 2.3 there is a g in Γ b suchthat ( g · Y a ) ∩ Y a has positive measure and in particular is nonempty.This contradicts our observation that if φ ( y ) > / g is in Γ b , then φ ( g · y ) < /
2. Similarly, if Y b has positive measure, one obtains acontradiction by finding a g in Γ a such that g · Y b intersects Y b . It mustbe, therefore, that the orbit equivalence relation is nonamenable. (cid:3) We finish this section with a simple but powerful observation ofMonod.
Example 4.9. [17] If A is a countable dense subring of R , let H ( A ) denote the group consisting of all orientation preserving homeomor-phisms of P ( R ) which fix the point at infinity and which are piecewise PSL ( A ) . Suppose that α is in PSL ( A ) and that α does not fix ∞ .As a fractional linear transformation, the graph of α is a hyperbola.If r ∈ R is sufficiently large in magnitude, then α ( t ) = t + r has twosolutions a < b ; set α r ( t ) = ( α ( t ) if a < t < bt + r otherwise . If r is moreover an integer, then α r is in H ( A ) . It follows that therestriction of the orbit equivalence relation of PSL ( A ) to R coincideswith the corresponding restriction of the orbit equivalence relation of H ( A ) ’s action on R . Thus, by Theorem 3.5 and the results of [9] mentioned above, H ( A ) is nonamenable whenever A is a dense subringof R . Example 4.10. [14]
Let P ( Z ) denote the subgroup of H ( Z ) consistingof those elements which have a continuous derivative. By unpublishedwork of Thurston, P ( Z ) is isomorphic to Richard Thompson’s group F . In fact α ( t ) = t + 1 β ( t ) = t if t ≤ t − t if ≤ t ≤ − t if ≤ t ≤ t + 1 if ≤ t is the standard set of generators with respect to the usual finite presen-tation of F (see [14] ). It is not difficult to see that the orbit equivalencerelation of P ( Z ) ’s action on P ( R ) coincides with that of PSL ( Z ) ex-cept for the point at infinity. Since PSL ( Z ) ∪ { t t + 1 / } generates PSL ( Z [1 / , it follows that h t t/ , β i is nonamenable. The previous example is less relevant to the amenability problem for F , however, than it might initially appear. For instance, Lodha hasshown that if Γ is any subgroup of H ( R ) which is isomorphic to F ,then the orbit equivalence relation of Γ’s action on R is λ -amenablewhere λ is Lebesgue measure. It is unclear whether this is true if Γ isonly assumed to be a subgroup of the homeomorphism group of R . MENABLE EQUIVALENCE RELATIONS 11 Closure properties of amenable equivalence relations
One of the most basic facts about amenable groups is that they areclosed under taking subgroups, extensions, and directed unions. Theseoperations have their analogs in the setting of countable Borel equiva-lence relations as well. Notice that if H ≤ G , then E HX ⊆ E GX whenever G acts on a standard Borel space. Also, if G is an increasing union ofa sequence of subgroups G n ( n < ∞ ), then E GX = S n E G n X . Since theproperty of being µ -hyperfinite is clearly inherited to subequivalencerelations, we have the following corollary of Theorem 3.5. Proposition 5.1. If E is a subequivalence relation of a µ -amenableequivalence relation is µ -amenable. While there is no natural notion of extension in the setting of equiv-alence relations, it is easy to formulate what is meant by a productof equivalence relations. It is straightforward to verify the followinganalog of the closure of the class of amenable groups under takingproducts.
Proposition 5.2.
Products of µ -amenable equivalence relations areamenable with respect to the corresponding product measure. The following is the corresponding analog of the amenability of in-creasing unions of amenable groups.
Theorem 5.3. [5] [13]
Suppose that E n ( n < ∞ ) is an increasingsequence of countable Borel equivalence relations on a standard Borelspace X . If µ is a standard measure on X and each E n is µ -hyperfinite,then S n E n is µ -hyperfinite. The power of the closure properties mentioned in this section is thatthey afford some flexibility which has no analog in the algebraic setting.For instance, while the equivalence relations E n in the previous theoremare required to be nested, they need not come from a nested sequence ofgroups. It is also sometimes fruitful to generate equivalence relationswith partial homeomorphisms rather than full automorphisms of anunderlying space. Example 5.4.
Consider the following homeomorphisms of P ( R ) = R ∪{∞} : α ( t ) = t +1 / and β ( t ) = − /t . For < r < ∞ , define α r tobe the restriction of α to [ − r, − /r ] . Let E be the equivalence relationgenerated by α and β and E r be the equivalence relation generated by β and α r (i.e. E r is the smallest equivalence relation such that for all t , ( t, t + 1 / ∈ E r and if additionally − r ≤ t ≤ − /r , then ( t, − /t ) ∈ E r ). Notice that if < r < s < ∞ , then E r ⊆ E s ⊆ E and that E = S r> E r . Thus there is an r such that < r < ∞ such that E r is nonamenable. In fact a more careful analysis reveals that E = E ,although this is not relevant for the point we wish to illustrate here. Example 5.5.
Suppose that α and β are homeomorphisms such thatthe action of h α, β i on R generates a nonamenable equivalence relation.Suppose further that α n ( n < ∞ ) and β n ( n < ∞ ) are sequences ofhomeomorphisms such that for all but countably many t , α n ( t ) = α ( t ) and β n ( t ) = β ( t ) holds for all but finitely many n . It follows thatthere exists an n such that the action of h α n , β n i on R generates anonamenable equivalence relation. To see this, let X n denote the setof all t in R such that for all k ≥ n , α k ( t ) = α ( t ) and β k ( t ) = β ( t ) .Define E n to be the equivalence relation generated by the restrictions of α n and β n to X n . It follows that E n ( n < ∞ ) is an increasing sequenceof countable Borel equivalence relations which, off a countable subset of R , unions to the equivalence relation generated by α and β . The claimnow follows from Theorem 5.3. References [1] E. Breuillard and T. Gelander. On dense free subgroups of Lie groups.
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Department of Mathematics, Cornell University, Ithaca, NY 14853–4201, USA
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