The space of stable weak equivalence classes of measure-preserving actions
aa r X i v : . [ m a t h . D S ] O c t The space of stable weak equivalence classes ofmeasure-preserving actions
Lewis Bowen ∗ and Robin Tucker-Drob † October 11, 2018
Abstract
The concept of (stable) weak containment for measure-preserving actions of a count-able group Γ is analogous to the classical notion of (stable) weak containment of unitaryrepresentations. If Γ is amenable then the Rokhlin lemma shows that all essentiallyfree actions are weakly equivalent. However if Γ is non-amenable then there can bemany different weak and stable weak equivalence classes. Our main result is that theset of stable weak equivalence classes naturally admits the structure of a Choquet sim-plex. For example, when Γ = Z this simplex has only a countable set of extreme pointsbut when Γ is a nonamenable free group, this simplex is the Poulsen simplex. Wealso show that when Γ contains a nonabelian free group, this simplex has uncountablymany strongly ergodic essentially free extreme points. Keywords : weak containment, pmp actions
MSC :37A35
Contents ∗ supported in part by NSF grant DMS-1500389, NSF CAREER Award DMS-0954606 † supported in part by NSF grant DMS-1600904 Preliminaries 7
A. Kechris introduced the notion of weak containment for group actions as an analogue ofweak containment for unitary representations [Kec10, II.10 (C)]. Given a countable groupΓ and probability measure-preserving (pmp) actions a := Γ y a ( X, µ ) , b := Γ y b ( Y, ν ) onstandard probability spaces, we say a is weakly contained in b (denoted a ≺ b ) if for everyfinite measurable partition { P i } ni =1 of X , finite S ⊆ Γ and ǫ > { Q i } ni =1 of Y satisfying | µ ( γ a P i ∩ P j ) − ν ( γ b Q i ∩ Q j ) | < ǫ for all γ ∈ S and 1 ≤ i, j ≤ n (where the action of Γ y a X is denoted γ a x for γ ∈ Γ , x ∈ X for example). We say a is weakly equivalent to b , denoted a ∼ b , if both a ≺ b and b ≺ a .The Rokhlin Lemma is essentially equivalent to the statement that for the group Γ = Z all essentially free pmp actions are weakly equivalent. Indeed, as remarked in [Kec12], thisstatement holds for all countable amenable groups. However it fails for nonamenable groupsbecause strong ergodicity is an invariant of weak equivalence [Kec10, Prop. 10.6]. Thismotivates the problem of providing a description of the set of all weak equivalence classes,denoted by W Γ , for a given group Γ.We start with an equivalent definition of weak containment. Let Cantor denote anyspace homeomorphic to a Cantor set. Let Γ act on
Cantor Γ by ( γx )( f ) = x ( γ − f ). LetProb Γ ( Cantor Γ ) denote the space of all Γ-invariant Borel probability measures on Cantor Γ equipped with the weak* topology. It is well-known that Prob Γ ( Cantor Γ ) is a Choquetsimplex : this means it is a compact convex subset of a locally convex topological vectorspace with the property that every element µ ∈ Prob Γ ( Cantor Γ ) can be uniquely writtenas a convex integral of extreme points of Prob Γ ( Cantor Γ ).Given an action a := Γ y a ( X, µ ), let Factor( a ) ⊆ Prob Γ ( Cantor Γ ) denote the set ofmeasures of the form Φ ∗ µ where Φ : X → Cantor Γ is a Γ-equivariant measurable mapand Φ ∗ µ = µ ◦ Φ − . The weak* closure of Factor( a ) is denoted W ( a ). It follows from[AW13] that a ≺ b if and only if W ( a ) ⊆ W ( b ) (see also [TD15, Prop. 3.6]). So the map a W ( a ) induces an injective map from the set of weak equivalence classes into the setof closed subsets of Prob Γ ( Cantor Γ ). We equip the latter with the Vietoris topology, and W Γ with the subspace topology. This topology, considered in [TD15], is a reformulation ofa construction due to Abert-Elek. The main result of [AE11] is that W Γ is compact (analternative proof is given in [TD15]).This motivates the question: what sort of subsets of Prob Γ ( Cantor Γ ) can have the form W ( a )? This is addressed in [AW13]: if a is strongly ergodic then W ( a ) is contained in the An action is essentially free if almost every point has trivial stabilizer. Γ ( Cantor Γ ). If a is ergodic but not strongly ergodic then W ( a )is a subsimplex of Prob Γ ( Cantor Γ ): that is, it is the convex hull of the extreme points ofProb Γ ( Cantor Γ ) contained in W ( a ). See Theorem 5.1 below.We now turn towards a description of stable weak equivalence classes where we obtain amore complete picture. We say that a is stably weakly contained in b , denoted a ≺ s b ,if a × i ≺ b × i where i denotes the trivial action of Γ on the unit interval equipped withLebesgue measure. If both a ≺ s b and b ≺ s a then we say the two actions are stablyweakly equivalent and denote this by a ∼ s b . Let SW ( a ) := W ( a × i ); by [TD15,Theorem 1.1] SW ( a ) is the closed convex hull of W ( a ) (see Lemma 5.2). Then a ≺ s b ifand only of SW ( a ) ⊆ SW ( b ). So a SW ( a ) induces an injective map from the set ofstable weak equivalence classes into the set of closed convex subsets of Prob Γ ( Cantor Γ ). Wedenote the set of stable weak equivalence classes with the induced topology by SW Γ . Likethe weak equivalence case, SW Γ is compact . By Theorem 5.1, if a is ergodic then SW ( a )is a subsimplex of Prob Γ ( Cantor Γ ).To simplify notation, let P := Prob Γ ( Cantor Γ ) and Closed( P ) denote the space ofall closed subsets of P equipped with the Vietoris topology, and let CloCon( P ) denote thecollection of all closed convex subsets of P . The space CloCon( P ) is compact, and it admitsa natural convex structure: if F , F ∈ CloCon( P ) and t ∈ [0 ,
1] then tF + (1 − t ) F ∈ CloCon( P ) is defined to be the set of all measures of the form tµ + (1 − t ) µ with µ i ∈ F i ( i = 1 , SW Γ is then a closed convex subset of CloCon( P ). Our main result isthat SW Γ is a Choquet simplex (Theorem 10.1). This means that for every α ∈ SW Γ thereexists a unique probability measure on the set of extreme points of SW Γ such that α is thebarycenter of this measure.Can we identify the simplex SW Γ up to affine homeomorphism? To begin answering thisquestion we need the following concept. An invariant random subgroup is a randomsubgroup of Γ whose law is invariant under conjugation. Let IRS(Γ) denote the space of allconjugation-invariant Borel probability measures on the space of subgroups of Γ. To any This can be proven in a manner similar to the case of W Γ . Alternatively, by Lemma 5.2 one can view SW Γ as the subspace of convex elements of W Γ . Because convexity is a closed property, SW Γ is closed in W Γ and therefore is compact. a = Γ y a ( X, µ ) we associate the element IRS( a ) defined byIRS( a ) := Stab ∗ µ where Stab : X → Sub(Γ) is the map Stab( x ) = { g ∈ Γ : g a x = x } and Sub(Γ) is the spaceof subgroups of Γ with the pointwise convergence topology. By [AE11] and [TD15], if a ∼ s b then IRS( a ) = IRS( b ). So we have a well-defined map IRS : SW Γ → IRS(Γ). In [TD15,Theorem 5.2] and [Bur15, Corollary 5.1] it is shown that this map is affine and continuous.It is also surjective by [AGV14, Proposition 45]. In [TD15] (see the remark after [TD15,Theorem 1.8]), it is shown that when Γ is amenable, IRS is a homeomorphism. So we havea complete description of SW Γ in the case where Γ is amenable.When Γ is nonamenable however, there can be many stable weak equivalence classeswhich map to a given IRS of Γ. If Γ is a nonamenable free group, then P. Burton showedthat the subsimplex of SW Γ consisting of all stable weak equivalence class of free actions, isa Poulsen simplex [Bur15]. This means that its extreme points are dense. There is a uniquePoulsen simplex up to affine homeomorphism [LOS78]. If Γ has property (T), then Theorem11.1 below shows that SW Γ is a Bauer simplex which means that the extreme points form aclosed subset of SW Γ . In particular, SW Γ cannot be a Poulsen simplex.In case Γ has a nonamenable free subgroup, Theorem 12.3 below shows that SW Γ has anuncountable set { S p } p ≥ of extreme points indexed by the interval [2 , ∞ ). Moreover, each S p is the class of a free, mixing, strongly ergodic action. The proof uses Okayasu’s resultthat the universal ℓ p (Γ)-representations of the free group are pairwise weakly inequivalent[Oka14]. Burton and Kechris have written a very recent survey article on weak containment [BK16].For every countable group Γ there exists a pmp action a such that all pmp actions of Γare weakly contained in a . This is known as the weak Rokhlin property [GTW06]. Thisproperty was introduced by Glasner-King where it was shown to imply a correspondencebetween generic properties of pmp actions and invariant measures [GK98].Moreover, every essentially free action weakly contains every Bernoulli action [AW13].This latter fact has been used to show that the cost of essentially free actions of Γ is maxi-5ized by the Bernoulli actions. Moreover, certain combinatorial quantities such as indepen-dence number of actions are weak equivalence invariants which allows one to use compactnessto prove that their extreme values are realized [CKTD13]. This paper also establishes equiv-alent definitions of weak containment in terms of the space of all actions and ultraproductsof actions.A residually finite group Γ has property MD if every action is stably weakly containedin a profinite action of Γ. It is known that residually finite amenable groups, free groups,and fundamental groups of closed hyperbolic 3-manifolds have property MD [BTD13]. Thisproperty is a strengthening of Lubotsky-Shalom’s property FD which is defined similarlybut for unitary representations instead of pmp actions [LS04]. It is unknown whether thedirect product of two free groups has MD or FD.The main result of [AE12] is that, for strongly ergodic actions, weak containment of agiven finite action implies actual containment of the same action. They apply this to showthat certain groups such as free groups and linear property (T) groups, admit an uncountablefamily of non-weakly-equivalent essentially free ergodic actions [AE12]. Ioana and Tucker-Drob strengthened the main result of [AE12] by generalizing finite actions to distal actions.Consequently, the weak equivalence class of a strongly ergodic action remembers the weakisomorphism class of its maximal distal factor [ITD16].Aaserud and Popa introduced several variants of weak containment in the context of orbit-equivalence [AP15]. Ab´ert and Elek show in [AE11] that the invariant random subgroup(IRS) of an action is a weak equivalence invariant. Tucker-Drob showed in [TD15] thatactions within a given weak equivalence class are unclassifiable up to countable structures.Peter Burton showed in [Bur15] that the space of stable weak equivalence classes naturallyforms a convex compact subset of a Banach space and, when Γ is amenable, identifies thissimplex as the simplex of IRS’s. The proofs used some ideas from an earlier draft of thispaper. Acknowledgements . After obtaining the proof that the space of stable weak equiva-lence classes forms a simplex, we naturally wondered what simplex could it be. It seemednatural to guess that for the free group, one obtains a Poulsen simplex. Peter Burton’s In [BTD13] it was shown that fundamental groups of virtually fibered hyperbolic 3-manifolds haveproperty MD. By [Ago13] all closed hyperbolic 3-manifolds are virtually fibered. • An action Γ y ( X, µ ) is pmp if µ is a probability measure and the action is measure-preserving. • An action Γ y ( X, µ ) is essentially free if for a.e. x ∈ X , the stabilizer of x in Γ istrivial. Throughout this paper,
Cantor denotes the Cantor set, Γ a countable group, P := Prob Γ ( Cantor Γ )the space of invariant Borel probability measures on Cantor Γ equipped with the weak* topol-ogy, P erg ⊆ P the subspace of ergodic invariant measures, Closed( P ) the space of closedsubsets of P with the Vietoris topology, and CloCon( P ) the space of closed convex subsetsof P . Moreover, if a = Γ y a ( X, µ ) is a pmp action then Factor( a ) ⊆ P is the set of allmeasures of the form Φ ∗ µ where Φ : X → Cantor Γ is measurable and Γ-equivariant. Also W ( a ) is the weak* closure of Factor( a ) and SW ( a ) = W ( a × i ) where i denotes the trivialaction of Γ on the unit interval with respect to Lebesgue measure. We let W Γ ⊆ Closed( P )denote the collection of all closed subsets of the form W ( a ) and SW Γ ⊆ Closed( P ) denotesthe collection of all closed subsets of the form SW ( a ) over all pmp actions a of Γ. Note that SW Γ ⊆ CloCon( P ) by [TD15, Theorem 1.1].If a = Γ y a ( X, µ ) then the action of Γ on X is denoted g a x for g ∈ Γ , x ∈ X . For t > t a by t a = Γ y a ( X, tµ ). In other words, it is the same action, we simplyscale the measure by t . If b = Γ y b ( Y, ν ) is another action then we define a ⊕ b to be theaction a ⊕ b = Γ y a ⊕ b ( X ⊔ Y, µ ⊕ ν ) where X ⊔ Y denotes the disjoint union of X and Y , µ ⊕ ν ( E ) = µ ( E ∩ X ) + ν ( E ∩ Y ) for E ⊆ X ⊔ Y and g a ⊕ b x = g a x, g a ⊕ b y = g b y for x ∈ X , y ∈ Y and g ∈ Γ. 7
Strong ergodicity
Definition 1.
Let a = Γ y a ( X, µ ). We say that a sequence { B i } ∞ i =1 of measurable sets in X is asymptotically invariant (with respect to a ) if for every g ∈ Γ,lim i →∞ µ ( B i △ g a B i ) = 0 . We say that { B i } ∞ i =1 is nontrivial if lim sup i →∞ µ ( B i )(1 − µ ( B i )) >
0. The action a is strongly ergodic if it does not admit any nontrivial asymptotically invariant sequences.Equivalently, a is strongly ergodic if b ≺ a implies b is ergodic (see [CKTD13, Prop. 5.6]). Definition 2. If a and b are pmp actions of Γ and t ∈ [0 ,
1] then we write t b ≺ a to meanthat t b ⊕ (1 − t ) i ≺ a where i is the trivial action of Γ on a one point probability space.Since any pmp action trivially contains i , if c is any pmp action and s b ⊕ (1 − s ) c ≺ a forsome 0 < s ≤
1, then s b ≺ a .More generally, if a , b are any finite-measure-preserving actions then b ≺ a means that t b ≺ t a where t > t a is probability-measure-preserving.The main result of this section is: Theorem 3.1.
Let a be an ergodic but not strongly ergodic pmp action of Γ . Then for every < t < , t a ≺ a . The next result was obtained in [JS87, Proof of Lemma 2.3].
Lemma 3.2 (Asymptotically invariant sets are mixing) . Let a = Γ y a ( X, µ ) be ergodic andlet { B i } ∞ i =1 ⊆ X be an asymptotically invariant sequence with respect to a such that lim i →∞ µ ( B i ) = t for some < t < . If A , A are any measurable subsets of X then for every g ∈ Γ , lim i →∞ | µ ( B i ∩ A ∩ g a A ) − µ ( B i ) µ ( A ∩ g a A ) | = 0 . Corollary 3.3. If a = Γ y a ( X, µ ) is an ergodic but not strongly ergodic pmp action of Γ then for every t ∈ (0 , there exists an asymptotically invariant sequence { B i } such that lim i →∞ µ ( B i ) = t . roof. Let N ⊆ (0 ,
1) be the set of all numbers t ∈ (0 ,
1) such that there exists an asymptot-ically invariant sequence { B i } such that lim i →∞ µ ( B i ) = t . Suppose that { B i } and { C j } areasymptotically invariant sequences. Then { X \ B i } , { B i ∩ C i } and { B i ∪ C i } are asymptoti-cally invariant. From the previous lemma it follows that { − t, st, s + t − st : s, t ∈ N } ⊆ N .Since N is closed and nonempty, it follows that N = (0 ,
1) as claimed.
Proof of Theorem 3.1.
Let a = Γ y a ( X, µ ), P = { P , . . . , P k } be a finite Borel partition of X and 0 < t <
1. By the previous corollary there exists an asymptotically invariant sequence { B n } with lim n →∞ µ ( B n ) = t . By Lemma 3.2,lim n →∞ | µ ( B n ∩ P i ∩ g a P j ) − tµ ( P i ∩ g a P j ) | = 0for all P i , P j ∈ P and g ∈ Γ. Set Q ( n ) i = B n ∩ P i . The asymptotic invariance of { B n } andthe previous limit implieslim n →∞ | µ ( Q ( n ) i ∩ g a Q ( n ) j ) − tµ ( P i ∩ g a P j ) | = 0for any i, j and g ∈ Γ. This implies the theorem.
The main purpose of this section is to prove:
Theorem 4.1.
Let a = Γ y a ( X, µ ) , b = Γ y b ( Y, ν ) , and c = Γ y c ( Y ′ , ν ′ ) be pmp actions of Γ . Let us assume a is ergodic.1. If a ≺ s b ⊕ (1 − s ) c for some < s ≤ then a ≺ b . Moreover a is weakly containedin almost every ergodic component of b .2. If s b ⊕ (1 − s ) c ≺ a for some < s ≤ then b ≺ a . Moreover, almost every ergodiccomponent of b is weakly contained in a .3. If s b ⊕ (1 − s ) c ∼ a for some < s ≤ then b ∼ a . Moreover almost every ergodiccomponent of b is weakly equivalent to a . Part (1) is equivalent to [TD15, Theorem 3.12]. Part (3) follows from parts (1) and (2).So we need only prove part (2). We will need measure algebras as defined next.9 efinition 3 (Measure algebras) . Let (
X, µ ) denote a measure-space. Given measurablesets
A, B ⊆ X we say that A and B are µ -equivalent if µ ( A △ B ) = 0. Let A µ denote the µ -equivalence class of A . The measure-algebra of µ , denoted MALG µ , is the set of allclasses A µ where A ⊆ X is a measurable set of finite measure. We usually abuse notationby treating an element of MALG µ as if it were a subset of X instead of an equivalence class.The set MALG µ has a natural metric given by symmetric difference: the distance between A, B ∈ MALG µ is µ ( A △ B ). Note that if µ is a standard σ -finite measure then MALG µ isseparable; it contains a countable dense subset.We need the next few lemmas before proving the second statement of Theorem 4.1. Lemma 4.2.
Let a = Γ y a ( X, µ ) and b = Γ y b ( Y, ν ) be pmp actions of Γ and let t ∈ [0 , .Suppose that a is ergodic and that t b ≺ a . Then given any Borel partition B , . . . , B m − of Y , finite subset F ⊆ Γ , ǫ > and finite subset A ⊆ MALG µ , there exist B ′ , . . . , B ′ m − ⊆ X such that, letting B ′ = S j
0. Since t b ≺ a , for each n ∈ N we may find subsets B ( n )0 , . . . , B ( n ) m − ⊆ X such that, letting B ( n ) = S j Let a = Γ y a ( X, µ ) and b = Γ y b ( Y, ν ) be pmp actions of Γ . Let t ∈ [0 , .Suppose that a is ergodic and t b ≺ a . Then t b ⊕ (1 − t ) a ≺ a .Proof. Given Borel partitions { B , . . . , B m − } of Y and { A , . . . , A n − } of X along with afinite subset F ⊆ Γ and ǫ > B ′ , . . . , B ′ m − , A ′ , . . . , A ′ n − ⊆ X such that(i) A ′ i ∩ B ′ j = ∅ for all i < n and j < m ;(ii) P i,j Assume that s b ⊕ (1 − s ) c ≺ a for some 0 < s ≤ 1. Thisimmediately implies s b ≺ a . Let r n = P nk =0 s (1 − s ) k . We show by induction on n ≥ r n b ≺ a . We have r b = s b ≺ a by hypothesis. Assume for induction that r n b ≺ a . Then r n +1 b = ( s + (1 − s ) r n ) b ≺ s b ⊕ (1 − s )( r n b ) ≺ s b ⊕ (1 − s ) a ≺ a r n b ≺ a for all n andlim n r n = s P ∞ k =0 (1 − s ) k = 1 it follows that b ≺ a .Next we assume that b ≺ a . Let ν = R z ∈ Z ν z dη be the disintegration of ν correspondingto the ergodic decomposition of b , and for each z ∈ Z let b z = Γ y b ( Y, ν z ). We must showthat b z ≺ a almost surely. Let C = { z ∈ Z : b z a } . Suppose toward a contradiction that η ( C ) > 0. Let B be a countable Boolean algebra which generates the Borel sigma algebra on Y . For each finite subset Q ⊆ B , consider the space [0 , Q × Q of all functions δ : Q × Q → [0 , Q ⊆ [0 , Q × Q .Let I denote the set of all quadruples ( F, Q , δ, ǫ ) where F ⊆ Γ is finite, Q ⊆ B is finite, δ : F × Q × Q → [0 , 1] is such that δ ( g, · , · ) ∈ ∆ Q for all g ∈ F and ǫ ∈ (0 , ∩ Q . Let I ⊆ I denote the subset consisting of all ( F, Q , δ, ǫ ) ∈ I for which there does not exist any function f : Q → MALG µ satisfying X B,B ′ ∈ Q | µ ( g a f ( B ) ∩ f ( B ′ )) − δ ( g, B, B ′ ) | ≤ ǫ for all g ∈ F . For each ( F, Q , δ, ǫ ) ∈ I define the set C F, Q ,δ,ǫ := ( z ∈ C : ∀ g ∈ F, X B,B ′ ∈ Q | ν z ( g b z B ∩ B ′ ) − δ ( g, B, B ′ ) | ≤ ǫ ) . It follows from the definitions that C = S ( F, Q ,δ,ǫ ) ∈ I C F, Q ,δ,ǫ . Since this is a countable unionand η ( C ) > η ( C F , Q ,δ ,ǫ ) = t > F , Q , δ , ǫ ) ∈ I .Let C = C F , Q ,δ ,ǫ and define b = Z z ∈ C b z dη C = Γ y b ( Y, ν C ) b = Z z ∈ Z \ C b z dη Z \ C = Γ y b ( Y, ν Z \ C )where η C is the normalized restriction of η to C and ν C = R z ν z dη C , and similarly for η Z \ C and ν Z \ C . Then a ≻ b ∼ = t b ⊕ (1 − t ) b . So by the first part of this proof we have a ≻ b .Since b ≺ a there exists some f : Q → MALG µ such that X B,B ′ ∈ Q | µ ( g a f ( B ) ∩ f ( B ′ )) − δ ( g, B, B ′ ) | ≤ ǫ for all g ∈ F which contradicts that ( F , Q , δ , ǫ ) ∈ I .12 Stable weak equivalence classes The purpose of this section is to prove: Theorem 5.1. If a is ergodic then SW ( a ) is a subsimplex of Prob Γ ( Cantor Γ ) . In otherwords, it is a closed convex subset whose extreme points are extreme points of Prob Γ ( Cantor Γ ) .Moreover, a is strongly ergodic if and only if W ( a ) is the set of extreme points of SW ( a ) ,in which case SW ( a ) is a Bauer simplex. If a is ergodic but not strongly ergodic then SW ( a ) = W ( a ) is a Poulsen simplex. Lemma 5.2. For any pmp action a of Γ , SW ( a ) is the closed convex hull of W ( a ) .Proof. We may assume a = Γ y a ( X, µ a ). We first show that SW ( a ) contains the closedconvex hull of W ( a ). So let t , . . . , t n > P i t i = 1 and µ , . . . , µ n ∈ W ( a ). It sufficesto show that P i t i µ i ∈ SW ( a ). By definition there exist factor maps ϕ ij : X → Cantor Γ such that lim j →∞ ϕ ij ∗ µ a = µ i for all i . Define Φ j : X × [0 , → Cantor Γ by Φ j ( x, t ) = ϕ ij ( x )if i is such that P k
Let a be an ergodic but not strongly ergodic action pmp action of Γ . Then W ( a ) = SW ( a ) .Proof. By Theorem 3.1 and Lemma 4.3, t a ⊕ (1 − t ) a ≺ a for any t ∈ (0 , ⊕ i t i a ≺ a for any sequence t , . . . , t n > P i t i = 1. In other words, W ( ⊕ i t i a ) ⊆ W ( a ). However, W ( ⊕ i t i a ) contains ⊕ i t i W ( a ) where the latter is defined to be13he collection of all measures of the form P i t i µ i with µ i ∈ W ( a ). Thus W ( a ) is convex.Lemma 5.2 now implies W ( a ) = SW ( a ). Proof of Theorem 5.1. To prove the first statement, suppose ν ∈ SW ( a ) ⊆ Prob Γ ( Cantor Γ )is not ergodic. So we can write it as ν = tν + (1 − t ) ν for some ν , ν ∈ Prob Γ ( Cantor Γ )such that ν and ν are mutually singular and t ∈ (0 , ν , ν ∈ SW ( a ). Therefore, ν cannot be an extreme point of SW ( a ). Thisproves that all extreme points of SW ( a ) are extreme points of Prob Γ ( Cantor Γ ).If a is strongly ergodic then it follows immediately that every measure in W ( a ) is ergodicand therefore extreme in Prob Γ ( Cantor Γ ). Since SW ( a ) is the closed convex hull of W ( a )this handles this case.Now suppose a is ergodic but not strongly ergodic. To see that the extreme points aredense, observe that every measure in Factor( a ) is ergodic (hence extreme) and SW ( a ) = W ( a ) is the weak* closure of Factor( a ) by Lemma 5.3. For simplicity, in this section we let P = Prob Γ ( Cantor Γ ). This is a compact metrizablespace in the weak* topology. Let Closed( P ) be the space of all closed subsets of P withthe Vietoris topology with respect to which Closed( P ) is a compact metrizable space. Let W Γ := { W ( a ) } a ⊆ Closed( P ) and SW Γ := { SW ( a ) } a ⊆ Closed( P ). In [TD15] it isproven that the topologies induced on W Γ and SW Γ from their inclusions into Closed( P )are equivalent to the topologies defined in [AE11] (another proof is in [Bur15, Theorem 3.1]).The next theorem is the main result of [AE11]: Theorem 6.1. Both W Γ and SW Γ are closed subsets of Closed( P ) . Therefore, W Γ and SW Γ are compact metrizable spaces. For each ρ ∈ P let W ( ρ ) := W ( a ) ⊆ P where a = Γ y ( Cantor Γ , ρ ). Similarly, let SW ( ρ ) := SW ( a ). We will frequently make use of the following facts:141) For every pmp action a = Γ y a ( X, µ ) there is a measure η ∈ P such that Γ y ( Cantor Γ , η )is isomorphic to a .(2) For any two pmp actions a = Γ y a ( X , µ ) and a = Γ y a ( X , µ ) of Γ, thereare measures η , η ∈ P whose supports are disjoint such that Γ y ( Cantor Γ , η ) isisomorphic to a and Γ y ( Cantor Γ , η ) is isomorphic to a Clearly (1) follows from (2). To see (2), let C and C be nonempty disjoint clopen subsetsof Cantor and for i = 0 , 1, let ϕ i : X i → C i be injections, and define Φ i : X i → Cantor Γ by Φ i ( x )( g ) = ϕ i (( g − ) a i x ). Then Φ i is injective and equivariant, and the supports of η i := (Φ i ) ∗ µ i are contained in C Γ i , so the measures η , η work.We introduce some notation which will be useful throughout the rest of the paper. Notation . To ease notation, we will not distinguish between a measure µ ∈ P and thecorresponding action Γ y ( Cantor Γ , µ ). For example, we will say that a measure µ ∈ P is ergodic or essentially free if the corresponding action is. Similarly if ρ , ρ ∈ P we willwrite ρ ≺ ρ to mean that the action corresponding to ρ is weakly contained in the actioncorresponding to ρ . As a corollary to Theorem 6.1, we will show that SW is lower semi-continuous as a mapfrom P to SW Γ . In general, if C , C , . . . ⊆ P are closed subsets then we define lim inf i C i to be the set of all µ ∞ ∈ P such that there exist µ i ∈ C i (for i ∈ N ) such that lim i µ i = µ ∞ . Corollary 6.2. [ SW is lower semi-continuous] If { µ i } i is a sequence in P and lim i µ i = µ ∞ then SW ( µ ∞ ) ⊆ lim inf i SW ( µ i ) . Remark . SW is not continuous in general. For example, consider the case when Γ = Z . Itis possible to find a sequence of measures µ i ∈ P such that Γ y ( Cantor Γ , µ i ) is essentiallyfree for all i but lim i µ i = δ x is the Dirac measure on a fixed point x ∈ Cantor Γ . By theRokhlin Lemma, SW ( µ i ) = P for all i and SW ( µ i ) = SW ( δ x ) since SW ( δ x ) is the subspaceof measures supported on fixed points. 15 roof. Since SW Γ is compact, after passing to a subsequence, we may assume that lim i SW ( µ i ) = SW ( ν ) for some ν ∈ P . Since µ ∞ = lim i µ i it follows that µ ∞ ∈ SW ( ν ). Thus µ ∞ ≺ s ν and therefore SW ( µ ∞ ) ⊆ SW ( ν ). Let P erg denote the extreme points of P = Prob Γ ( Cantor Γ ). Let Prob( P erg ) denote thespace of Borel probability measures on P erg . Let π : Cantor Γ → P erg be an ergodicdecomposition map. By definition this means that π is a Γ-invariant Borel map satisfying • For each e ∈ P erg , e ( { x ∈ Cantor Γ : π ( x ) = e } ) = 1. • For each µ ∈ P , µ = R e ∈ P erg e d π ∗ ( µ ).Furthermore, π is unique in the following sense: if π ′ is another such map then the set { x : π ( x ) = π ′ ( x ) } is µ -null for all µ ∈ P [GS00].Let π ∗ : P → Prob( P erg ) be the associated affine map which takes a measure µ ∈ P toits ergodic decomposition π ∗ ( µ ) ∈ Prob( P erg ). In what follows we will abuse notation andwrite π ( µ ) for π ∗ ( µ ). If κ ∈ Prob( P ) then we let β ( κ ) ∈ P denote the Barycenter of κ . Bydefinition, β ( κ ) = Z P µ dκ ( µ ) . So β ( π ( µ )) = µ , and if κ ∈ Prob( P erg ) then π ( β ( κ )) = κ . Definition 4. Let ( X, A , µ ) and ( Y, B , ν ) be probability spaces. A coupling of µ with ν isa probability measure ρ on ( X × Y, A ⊗ B ) such that (proj X ) ∗ ρ = µ and (proj Y ) ∗ ρ = ν .Let ( Z, C , η ) be another probability space and let ρ be a coupling of µ with ν , and let σ be a coupling of ν with η . Then the composition of ρ and σ , denoted ρ ◦ σ , is the couplingof µ with η defined by ρ ◦ σ = R Y ρ y × σ y dν , where ρ = R Y ρ y × δ y dν and σ = R Y δ y × σ y dν are the respective disintegrations of ρ and σ via the natural projection maps. Lemma 7.1. Let λ and ω be Borel probability measures on P and assume that there is acoupling ρ of λ with ω which concentrates on the set { ( µ, ν ) : µ ≺ s ν } . Assume in additionthat there is a ω -conull set P ω ⊆ P such that the measures in P ω are mutually singular.Then β ( λ ) ≺ s β ( ω ) . 16e note that the hypothesis on ω is automatically satisfied if ω concentrates on P erg . Proof. Let ρ = R P ω ρ ν × δ ν dω ( ν ) be the disintegration of ρ over ω . Then for ω -a.e. ν ,the measure ρ ν concentrates on SW ( ν ), hence β ( ρ ν ) ∈ SW ( ν ), since SW ( ν ) is a closedconvex set. We have β ( λ ) = R β ( ρ ν ) dω ( ν ). Fix an atomless Borel probability measure ν on Cantor . Also, let Γ act on Cantor Γ × Cantor by g ( x, y ) = ( gx, y ) . Then β ( ρ ν ) ≺ ν × ν for ω -a.e. ν . Fix a Borel partition P = { P , . . . , P k } of Cantor Γ , ǫ > F ⊆ Γ. It suffices to show there exists a Borel partition { U , . . . , U k } of Cantor Γ × Cantor such that (cid:12)(cid:12)(cid:12)(cid:12)Z P β ( ρ ν )( P i ∩ gP j ) − ν × ν ( U i ∩ gU j ) dω ( ν ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ for every g ∈ F and 1 ≤ i, j ≤ k .Let { Q ( n ) } ∞ n =1 be an enumeration of all clopen partitions of Cantor Γ × Cantor of theform Q ( n ) = { Q ( n )1 , . . . , Q ( n ) k } . There are only countably many such partitions. For ω -a.e. ν , since β ( ρ ν ) ≺ s ν × ν , and because the clopen sets are dense in the measure algebra of ν × ν , there exists some number n ( ν ) ∈ N such that (cid:12)(cid:12)(cid:12) β ( ρ ν )( P i ∩ gP j ) − ν × ν (cid:16) Q ( n ( ν )) i ∩ gQ ( n ( ν )) j (cid:17)(cid:12)(cid:12)(cid:12) < ǫ. for every g ∈ F and 1 ≤ i, j ≤ k . We choose n ( ν ) to be the smallest natural number withthis property. With this choice, the map ν n ( ν ) is measurable.Let M denote the set of all m ∈ N such that ω ( { ν ∈ P : n ( ν ) = m } ) > . Define κ m = Z ν : n ( ν )= m ν × ν dω ( ν ) . This is a Γ-invariant Borel measure on Cantor Γ × Cantor . Moreover, the measures { κ m : m ∈ M } are mutually singular since the measures in the ω -conull set P ω are mutually singular.So there exists a Borel partition R = { R m } m ∈ M of Cantor Γ × Cantor such that κ m ( Cantor Γ × Cantor ) = κ m ( R m )17nd R m is Γ-invariant for all m ∈ M . Thus for ω -a.e. ν ∈ P we have ν × ν ( E ) = ν × ν ( E ∩ R n ( ν ) )for any Borel E ⊆ Cantor Γ × Cantor . Let U i = [ m ∈ M R m ∩ Q ( m ) i . Then { U , . . . , U k } is a Borel partition of Cantor Γ × Cantor and for any g ∈ F , 1 ≤ i, j ≤ k , (cid:12)(cid:12)(cid:12)(cid:12)Z β ( ρ ν )( P i ∩ gP j ) − ν × ν ( U i ∩ gU j ) dω ( ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | β ( ρ ν )( P i ∩ gP j ) − ν × ν ( U i ∩ gU j ) | dω ( ν )= X m ∈ M Z ν : n ( ν )= m | β ( ρ ν )( P i ∩ gP j ) − ν × ν ( U i ∩ gU j ) | dω ( ν )= X m ∈ M Z ν : n ( ν )= m (cid:12)(cid:12)(cid:12) β ( ρ ν )( P i ∩ gP j ) − ν × ν (cid:16) Q ( m ) i ∩ gQ ( m ) j (cid:17)(cid:12)(cid:12)(cid:12) dω ( ν ) ≤ ǫ. The main theorem of this section is: Theorem 8.1. [Coupling Theorem] Let µ, ν ∈ P .(i) µ ≺ s ν if and only if there exists a coupling ρ of π ( µ ) and π ( ν ) which concentrates onthe set { ( e , e ) ∈ P erg × P erg : e ≺ s e } (ii) µ ∼ s ν if and only if there exists a coupling ρ of π ( µ ) and π ( ν ) which concentrates onthe set { ( e , e ) ∈ P erg × P erg : e ∼ s e } Moreover, if µ ∼ s ν and ρ is any coupling of π ( µ ) and π ( ν ) which concentrates on { ( e , e ) ∈ P erg × P erg : e ≺ s e } , then ρ in fact concentrates on { ( e , e ) ∈ P erg × P erg : e ∼ s e } . SW Γ . Definition 5. To each open subset U of P we associate the sets B U = { ρ ∈ P : SW ( ρ ) ∩ U = ∅} C U = { ρ ∈ P : π ( ρ )( B U ) > } . The following proposition gives some basic properties of the sets B U and C U which willbe used several times below. Proposition 8.2. Let U and V be open subsets of P .(i) C U ∩ P erg = B U ∩ P erg .(ii) U ⊆ B U and U ∩ P erg ⊆ C U .(iii) If µ ∈ B U and µ ≺ s ν then ν ∈ B U .(iv) If U ⊆ V then B U ⊆ B V and C U ⊆ C V .(v) B U is open and C U is Borel.Proof. Statements (i) through (iv) all follow from the definitions. For (v), to see B U is openit suffices to show that P \ B U is closed. Assume ρ n ∈ P \ B U and ρ n → ρ ∈ P . Then SW ( ρ n ) ⊆ P \ U for all n , so lim inf n SW ( ρ n ) ⊆ P \ U since P \ U is closed. By Lemma6.2, SW ( ρ ) ⊆ lim inf n SW ( ρ n ) ⊆ P \ U , i.e., ρ ∈ P \ B U . The set C U is Borel since π and B U are both Borel. Lemma 8.3. Let V ⊆ P be open.(1) Let µ ∈ C V . Then for any e ∈ P erg \ B V there exists a neighborhood U of µ with e B U .(2) Let L ⊆ C V be compact, and let ν ∈ P . Then for any ǫ > there exists an open set U ⊆ P with L ⊆ U and π ( ν )( B U \ B V ) < ǫ .(3) Let λ be a Borel probability measure on P , and let ν ∈ P . Then for any ǫ > thereexists an open set U ⊆ P with λ ( C V \ U ) = 0 and π ( ν )( B U \ B V ) < ǫ . roof. (1): Assume toward a contradiction that there is some e ∈ P erg \ B V such that forall open neighborhoods U of µ we have e ∈ B U , i.e., SW ( e ) ∩ U = ∅ . This means that µ ∈ SW ( e ), so that µ ≺ s e and therefore π ( µ )( { e ′ ∈ P erg : e ′ ≺ s e } ) = 1 (1)by Theorem 4.1 (2). From (1) and the hypothesis µ ∈ C V we conclude that there is some e ′ ∈ P erg ∩ B V with e ′ ≺ s e . Therefore, by Proposition 8.2, e ∈ B V , a contradiction.(2): Fix ǫ > 0. Let { O n } n ∈ N be a countable basis of open subsets of P and let { U n } n ∈ N enumerate all finite unions of elements of { O n } n ∈ N . Claim 1. Let e ∈ P erg \ B V . Then there exists some n ∈ N such that L ⊆ U n and e B U n .Proof of Claim. By part (1), for each µ ∈ L there is some n ( µ ) ∈ N such that µ ∈ O n ( µ ) and e B O n ( µ ) . Then L ⊆ S µ ∈ L O n ( µ ) , and since L is compact there exists some finite Q ⊆ L such that L ⊆ S µ ∈ Q O n ( µ ) . Taking any n ∈ N with U n = S µ ∈ Q O n ( µ ) works since e S µ ∈ Q B O n ( µ ) = B U n . [Claim]For each e ∈ P erg \ B V let n ( e ) = min { n ∈ N : L ⊆ U n and e B U n } . Let N be solarge that π ( ν )( { e ∈ P erg \ B V : n ( e ) < N } ) > π ( ν )( P erg \ B V ) − ǫ, and define U = T { U n : n < N and L ⊆ U n } . Then U is open and L ⊆ U . Furthermore, P erg ∩ B U \ B V ⊆ { e ∈ P erg \ B V : n ( e ) ≥ N } since if e B V is such that n ( e ) < N , then U ⊆ U n ( e ) and therefore e B U (since e B U n ( e ) ). This shows that π ( ν )( B U \ B V ) < ǫ .(3): The measure λ is regular, so we may find a sequence L , L , . . . , of compact subsetsof C V with λ ( C V \ L n ) → 0. For each n apply (2) to find an open U n with L n ⊆ U n and π ( ν )( B U n \ B V ) < ǫ/ n . Let U = S n U n . Then λ ( C V \ U ) = 0, and B U = S n B U n , hence π ( ν )( B U \ B V ) < ǫ . Let U denote a nonprincipal ultrafilter on N and ( X, µ ) be a standard Borel probabilityspace. Define an equivalence relation ∼ U on X N by { x i } ∼ U { y i } if and only if { n ∈ N : x n = y n } ∈ U . Let X U := X N / ∼ U denote the set of all ∼ U equivalence classes. If { B n } is a20equence of subsets of X then we let [ B n ] ⊆ X U denote the set of all equivalence classes ofthe form [ x n ] with { n ∈ N : x n ∈ B n } ∈ U . For each Borel B ⊆ X we also let [ B ] ⊆ X U denote the set [ B ] := { [ x n ] : { n : x n ∈ B } ∈ U } corresponding to the constant sequence.If B n ⊆ X is a sequence of Borel sets then we define µ U ([ B n ]) := lim n → U µ ( B n ). Thisfunction extends in a unique way to a probability measure, still denoted µ U , on the sigma-algebra B ( X U ) generated by all sets of the form [ B n ] where each B n ⊆ X is Borel. We let σ ( µ U ) denote the completion of B ( X U ) with respect to µ U . Thus ( X U , µ U ) (equipped withthe sigma algebra σ ( µ U )) is a probability space called the ultrapower of ( X, µ ). In general,it is not standard because the corresponding measure algebra need not be separable. See[CKTD13] for more details on ( X U , µ U ).There is a natural measure algebra embedding I : MALG µ ֒ → MALG µ U given by B µ [ B ] µ U . The map I preserves the algebra structure and it is continuous, hence it also preservesthe σ -algebra structure. If we assume that X is a compact Polish space, then the followingproposition shows that the limit map [ x n ] lim n → U x n , gives a natural point realization ofthe embedding I . Proposition 8.4. (1) Let K be a compact Polish space. Let ϕ n : X → K , n ∈ N , be asequence of Borel functions from X to K . Then the function ϕ : X U → K given by ϕ ([ x n ]) = lim n → U ϕ n ( x n ) is measurable.(2) Assume that X is a compact Polish space. Then the map lim U : X U → X , de-fined by lim U ([ x n ]) = lim n → U x n , is measurable, and for each Borel B ⊆ X we have µ U (lim − U ( B ) △ [ B ]) = 0 . In particular, lim U : ( X U , µ U ) → ( X, µ ) is measure preserving.Proof. For (1), let d be a compatible metric on K and fix an open set V ⊆ K . Since V is openwe have V = S m V m where V m = { k ∈ V : d ( k, K \ V ) > /m } . We then have the equality ϕ − ( V ) = [ ϕ − n ( V )] ∪ [ ϕ − n ( V )] ∪· · · , which shows ϕ is measurable. Statement (2) correspondsto the case X = K , ϕ n = id X for all n , and ϕ = lim U . In this case, using the notation fromabove, the sequence [ V ] , [ V ] , . . . increases to lim − U ( V ), and thus I ( V µm ) → lim − U ( V ) µ U inMALG µ U . But also I ( V µm ) → I ( V µ ) by continuity of I , hence I ( V µ ) = lim − U ( V ) µ U . Thus,the collection B , of all Borel subsets B ⊆ X satisfying lim − U ( B ) µ U = I ( B µ ), contains allopen subsets of X , and it is also a σ -algebra since the maps B lim − U ( B ) µ U and B I ( B µ )21oth preserve σ -algebra operations. This shows B contains every Borel set, and completesthe proof of (2). Proof of Theorem 8.1. (i): Assume first that there exists a coupling ρ of π ( µ ) and π ( ν ) as in(i). Then the disintegration of ρ with respect to the right projection map ( e , e ) e is ofthe form ρ = R ρ e × δ e d π ( ν )( e ). For π ( ν )-almost every e ∈ P erg the measure ρ e concentrateson { e ′ : e ′ ≺ s e } . Since SW ( e ) is convex (by Theorem 5.1), β ( ρ e ) ≺ s e . By Lemma 7.1, µ = β ( π ( µ )) = R β ( ρ e ) d π ( ν )( e ) ≺ s R e d π ( ν )( e ) = ν .Now assume that µ ≺ s ν . Let ν ′ ∈ P be such that Γ y ( Cantor Γ , ν ′ ) is isomorphic tothe product of Γ y ( Cantor Γ , ν ) with the identity action of Γ on ([0 , , Leb). Then there is acoupling σ of π ( ν ′ ) and π ( ν ) which concentrates on pairs of isomorphic ergodic components.If we can find a coupling ρ ′ of π ( µ ) and π ( ν ′ ) which concentrates on pairs ( e , e ) with e ≺ e , then the composition ρ = ρ ′ ◦ σ will be the desired coupling of π ( µ ) with π ( ν ).Therefore, after replacing ν by ν ′ if necessary, we may assume without loss of generality that SW ( ν ) = W ( ν ) so that in fact µ ≺ ν .Fix a non-principal ultrafilter U on N . Let ( Cantor Γ U , ν U ) denote the ultrapower of( Cantor Γ , ν ). As π ( ν ) is a measure on P which concentrates on P erg , the ultrapower π ( ν ) U is a measure on the space P U which concentrates on the set [ P erg ] which we identify with P erg U . By Proposition 8.4, we have that π ( ν ) = lim U ∗ π ( ν ) U . In particular, for π ( ν ) U -almostevery [ e n ] ∈ P erg U , we have lim n → U e n ∈ P erg . For each [ e n ] ∈ P erg U we let Q U [ e n ] denotethe measure on Cantor Γ U determined by Q U [ e n ]([ A n ]) = lim n → U e n ( A n ) for A n ⊆ Cantor Γ Borel.Since µ ≺ ν there exist Borel factor maps Φ n : Cantor Γ → Cantor Γ with (Φ n ) ∗ ν → µ .Let Φ : ( Cantor Γ ) U → Cantor Γ be the ultralimit function given by Φ([ x n ]) = lim n → U Φ n ( x n ).By [TD15, Proposition 3.11] we have Φ ∗ ( ν U ) = lim n → U (Φ n ) ∗ ν = µ and Φ ∗ Q U [ e n ] = lim n → U (Φ n ) ∗ e n for every [ e n ] ∈ P erg U . The map [ e n ] Φ ∗ Q U [ e n ] = lim n → U (Φ n ) ∗ e n , from P erg U to P , istherefore measurable by Proposition 8.4. By [TD15, Proposition A.1], the decomposition ν = R e d π ( ν )( e ) yields ν U = R Q U [ e n ] d π ( ν ) U ([ e n ]), and hence µ = Φ ∗ ( ν U ) = Z lim n → U (Φ n ) ∗ e n d π ( ν ) U ([ e n ]) . (2)22et ρ be the measure on P × P defined by ρ = Z π (cid:16) lim n → U (Φ n ) ∗ e n (cid:17) × δ lim n → U e n d π ( ν ) U ([ e n ]) . (3)Then ρ concentrates on P erg × P erg , and (2) and Proposition 8.4 show that ρ is a couplingof π ( µ ) and π ( ν ). Claim 2. Let V ⊆ P be open. Then ρ ( B V × ( P erg \ B V )) = 0 .Proof of Claim. Suppose not. Then the expression (3) implies that π ( ν ) U ( D ) > 0, where D = n [ e n ] ∈ P erg U : lim n → U (Φ n ) ∗ e n ∈ C V and lim n → U e n B V o . Let λ denote the push-forward of π ( ν ) U under the map [ e n ] lim n → U (Φ n ) ∗ e n , so that λ isa Borel probability measure on P . By Lemma 8.3.(3) we may find an open set U ⊆ P such that λ ( C V \ U ) = 0 and π ( ν )( B U \ B V ) < π ( ν ) U ( D ). Thus, for π ( ν ) U -almost every[ e n ] ∈ D we have lim n → U (Φ n ) ∗ e n ∈ U and (by Proposition 8.4) [ e n ] [ B V ]. Therefore, π ( ν ) U ( D ) ≤ π ( ν ) U ( D ), where D = n [ e n ] ∈ P erg U : lim n → U (Φ n ) ∗ e n ∈ U and [ e n ] [ B V ] o . Since π ( ν )( B U \ B V ) < π ( ν ) U ( D ) ≤ π ( ν ) U ( D ), the set D \ ([ B U ] \ [ B V ]) is ν U -non-nulland hence nonempty. Fix any [ e n ] in this set. Then [ e n ] [ B V ] (since [ e n ] ∈ D ), and[ e n ] [ B U ] \ [ B V ], hence [ e n ] [ B U ] . (4)On the other hand, [ e n ] ∈ D implies lim n → U (Φ n ) ∗ e n ∈ U . Since U is an open neighborhoodabout lim n → U (Φ n ) ∗ e n we have { n : (Φ n ) ∗ e n ∈ U } ∈ U . For each n with (Φ n ) ∗ e n ∈ U wehave (Φ n ) ∗ e n ∈ SW ( e n ) ∩ U and so e n ∈ B U . Therefore { n : e n ∈ B U } ∈ U , i.e., [ e n ] ∈ [ B U ],which contradicts (4). [Claim 2]Let { V i } i ∈ N be a countable base of open subsets of P . Then { ( e , e ) : e ≺ s e } = \ i ∈ N { ( e , e ) : e ∈ B V i ⇒ e ∈ B V i } , and ρ concentrates on this set by Claim 2. 23ii): If ρ is a coupling of π ( µ ) and π ( ν ) as in (ii), then in particular ρ ( { ( e , e ) : e ≺ s e } ) = 1, so µ ≺ s ν by part (i). Similarly, ν ≺ s µ , and thus ν ∼ s µ . The other direction of(ii) will follow from (i) once we establish the final statement of the theorem.Suppose µ ∼ s ν and let ρ be a coupling of π ( µ ) and π ( ν ) concentrating on { ( e , e ) : e ≺ s e } . Suppose toward a contradiction that ρ ( { ( e , e ) : e ≻ s e } ) < 1. Then thereexists an open subset U of P such that ρ ( { ( e , e ) : e B U and e ∈ B U } ) > , (5)where B U = { λ ∈ P : SW ( λ ) ∩ U = ∅} . The condition ρ ( { ( e , e ) : e ≺ s e } ) = 1 impliesthat ρ ( { ( e , e ) : e ∈ B U } ) = ρ ( { ( e , e ) : e ∈ B U and e ∈ B U } ) (6)Using (5) and (6) we compute π ( µ )( B U ) = ρ ( { ( e , e ) : e ∈ B U } )= ρ ( { ( e , e ) : e ∈ B U and e ∈ B U } ) < ρ ( { ( e , e ) : e ∈ B U and e ∈ B U } ) + ρ ( { ( e , e ) : e B U and e ∈ B U } )= ρ ( { ( e , e ) : e ∈ B U } )= π ( ν )( B U ) . On the other hand, since µ ≻ s ν , part (i) implies that we can find coupling e ρ of π ( µ ) and π ( ν ) such that e ρ ( { ( e , e ) : e ≻ s e } ) = 1 and therefore π ( ν )( B U ) = e ρ ( { ( e , e ) : e ∈ B U } ) ≤ e ρ ( { ( e , e ) : e ∈ B U } ) = π ( µ )( B U ) , a contradiction. The space CloCon( P ), of all closed convex subsets of P , is naturally endowed with a convexstructure: if F , F ∈ CloCon( P ) and 0 ≤ t ≤ tF + (1 − t ) F := { tµ + (1 − t ) µ : µ ∈ F , µ ∈ F } . , ω ) is a probability space and F : Ω → CloCon( P ) a measurablemap then Z F ( x ) dω ( x ) ⊆ P denotes the set of all measures in P of the form R σ ( x ) dω ( x ) where σ runs over all mea-surable σ : Ω → P satisfying σ ( x ) ∈ F ( x ) for ω -a.e. x . Theorem 9.1. Let ω be a Borel probability measure on P and assume that there is an ω -conull set P ω ⊆ P such that the measures in P ω are mutually singular. Then Z SW ( µ ) dω ( µ ) = SW (cid:18)Z µ dω ( µ ) (cid:19) . It follows that SW Γ is convex.Remark . Theorem 9.1 implies that SW ( t a ⊕ (1 − t ) b ) = tSW ( a ) + (1 − t ) SW ( b ) for allp.m.p. actions a and b of Γ, and all t ∈ [0 , µ a , µ b ∈ P of a and b respectively, whose supports are disjoint, and hence by Theorem 9.1 SW ( t a ⊕ (1 − t ) b ) = SW ( tµ a +(1 − t ) µ b ) = tSW ( µ a )+(1 − t ) SW ( µ b ) = tSW ( a )+(1 − t ) SW ( b ) . Proof. By Lemma 7.1, if f : P → P is a measurable map satisfying f ( µ ) ≺ s µ for ω -a.e. µ then Z f ( µ ) dω ( µ ) ∈ SW (cid:18)Z µ dω ( µ ) (cid:19) . This proves R SW ( µ ) dω ( µ ) ⊆ SW (cid:0)R µ dω ( µ ) (cid:1) . To prove the opposite containment, suppose that ν ∈ SW (cid:0)R µ dω ( µ ) (cid:1) . Then by Theorem8.1 there exists a coupling ρ of π ( ν ) and π ( R µ dω ) = π ( β ( ω )) such that ρ ( { ( e , e ) ∈ P erg × P erg : e ≺ s e } ) = 1 . Let ρ = R P erg ρ e × δ e d π ( β ( ω )) be the disintegration of ρ over π ( β ( ω )). Then β ( ρ e ) ≺ s e for π ( β ( ω ))-almost every e ∈ P erg , so after redefining ρ e on a π ( β ( ω ))-null set if necessary wemay assume without loss of generality that β ( ρ e ) ≺ s e for all e ∈ P erg . For each µ ∈ P let ρ µ := R ρ e d π ( µ )( e ). Then β ( ρ µ ) ≺ s β ( π ( µ )) = µ by Lemma 7.1. Since π ( β ( ω )) = R π ( µ ) dω we have Z ρ µ dω ( µ ) = Z Z ρ e d π ( µ )( e ) dω ( µ ) = Z ρ e d ( π ( β ( ω )))( e ) = π ( ν ) . µ β ( ρ µ ) witnesses that ν ∈ R SW ( µ ) dω ( µ ). This proves that R SW ( µ ) dω ( µ ) ⊇ SW (cid:0)R µ dω ( µ ) (cid:1) . To see that SW Γ is convex, given SW ( µ ) , SW ( ν ) ∈ SW Γ and t ∈ [0 , µ ′ and ν ′ , of µ and ν respectively, whose supports are disjoint. Then tSW ( µ ) + (1 − t ) SW ( ν ) = tSW ( µ ′ ) + (1 − t ) SW ( ν ′ ) = SW ( tµ ′ + (1 − t ) ν ′ ) ∈ SW Γ . 10 Simplex In this section, we prove SW Γ is a simplex. Let SW ext Γ ⊆ SW Γ denote the subspace ofextreme stable weak equivalence classes. More precisely, S ∈ SW ext Γ if and only if theequation S = tS + (1 − t ) S with S , S ∈ SW Γ and t ∈ (0 , 1) implies S = S = S . Theorem 10.1. For each stable weak equivalence class S ∈ SW Γ there exists a unique Borelprobability measure π ( S ) on SW ext Γ such that S = R E ∈ SW ext Γ E d π ( S ) . Furthermore, for any µ ∈ P we have π ( SW ( µ )) = SW ∗ π ( µ ) . Lemma 10.2. If S ∈ SW Γ is a subsimplex of P then it is extreme. In particular, if µ ∈ P erg then SW ( µ ) ∈ SW ext Γ . Conversely, if S ∈ SW ext Γ then there exist an ergodic µ ∈ P erg suchthat S = SW ( µ ) .Proof. Let S ∈ SW Γ be a subsimplex of P and suppose S = tS + (1 − t ) S for some S , S ∈ SW Γ and t ∈ (0 , ν ∈ S we must be able to write ν = tν + (1 − t ) ν for some ν i ∈ S i ( i = 1 , ν is ergodic, ν = ν = ν . So S ∩ P erg ⊆ S ∩ S . By hypothesis, S is the closed convex hull of S ∩ P erg . Since S and S are convex, S ⊆ S ∩ S . To obtain a contradiction, suppose ν ∈ S \ S . Let ν ∈ S .Then tν + (1 − t ) ν ∈ S . By the ergodic decomposition theorem, almost every ergodiccomponent of ν must be contained in S and therefore, ν ∈ S . This contradiction showsthat S ∩ S ⊆ S . So S = S = S as claimed.Suppose µ ∈ P erg . By Theorem 5.1, SW ( µ ) is a subsimplex of P . So the previousparagraph implies SW ( µ ) ∈ SW ext Γ .For the converse, suppose S ∈ SW ext Γ . Let µ ∈ P such that S = SW ( µ ). By Theorem9.1, S = SW ( µ ) = Z SW ( e ) d π ( µ )( e ) . SW ( µ ) is extreme, we must have SW ( e ) = S for π ( µ )-a.e. e ∈ P erg . Proof of Theorem 10.1. Lemma 10.2 shows that SW maps P erg onto SW ext Γ . So SW ∗ :Prob( P ) → Prob( SW Γ ) maps Prob( P erg ) onto Prob( SW ext Γ ). In addition, if µ ∈ P then SW ∗ π ( µ ) is a Borel probability measure on SW ext Γ whose barycenter is SW ( µ ) since Z SW ext Γ E dSW ∗ π ( µ )( E ) = Z SW ( e ) d π ( µ )( e ) = SW (cid:18)Z e d π ( µ )( e ) (cid:19) = SW ( µ )where the second equality holds by Theorem 9.1 and the other equalities hold by definition.This shows that every stable weak equivalence class is represented by a measure on SW ext Γ .We now show that this representation is unique.Let κ and κ be Borel probability measures on SW ext Γ with R E dκ ( E ) = S = R E dκ ( E ).We must show that κ = κ . By [Kec95, Theorem 18.1] and Lemma 10.2 there exists a uni-versally measurable map s : SW ext Γ → P erg with SW ( s ( E )) = E for all E ∈ SW ext Γ . For i ∈ { , } let µ i = β ( s ∗ κ i ) ∈ P . Then SW ( µ i ) = Z SW ( e ) ds ∗ κ i ( e ) = Z SW ( s ( E )) dκ i ( E ) = Z E dκ i ( E ) = S, so µ and µ are stably weakly equivalent. By Theorem 8.1 there exists a coupling ρ of π ( µ )and π ( µ ) with ρ ( { ( e , e ) : e ∼ s e } ) = 1. We have π ( µ i ) = π ( β ( s ∗ κ i )) = s ∗ κ i , so ρ is acoupling of s ∗ κ and s ∗ κ . Then ( SW × SW ) ∗ ρ is a coupling of κ and κ with( SW × SW ) ∗ ρ (cid:0)(cid:8) ( E , E ) ∈ ( SW ext Γ ) : E = E (cid:9)(cid:1) = ρ ( { ( e , e ) : e ∼ s e } ) = 1 . It follows that for any Borel B ⊆ SW ext Γ we have κ ( B ) = ( SW × SW ) ∗ ρ ( B × SW ext Γ ) =( SW × SW ) ∗ ρ ( B × B ) = ( SW × SW ) ∗ ρ ( SW ext Γ × B ) = κ ( B ) and so κ = κ .The second statement follows from the first and the fact that SW ( µ ) = R E dSW ∗ π ( µ )( E ).In [Bur15, Theorem 1.5], P. Burton shows that SW Γ is affinely homeomorphic to a convexcompact subset of a Banach space. The proof uses an abstract characterization of convexcompact subsets of Banach spaces due to Capraro and Fritz [CF13]. It now follows fromTheorem 10.1 that SW Γ is a Choquet simplex (equivalently, it is a convex compact subsetof a locally convex topological vector space with the property that every element admits aunique representation as the barycenter of a probability measure on the space of extremepoints). 27 Theorem 11.1. Suppose Γ is a countable group with property (T). Then SW Γ is a Bauersimplex; the set SW ext Γ ⊆ SW Γ of extreme points is closed.Proof. Let { S n } ⊆ SW ext Γ be a sequence of extreme stable weak equivalence classes. Supposelim n S n = S ∞ ∈ SW Γ . It suffices to show S ∞ is extreme.Because each S n is extreme, S n is a subsimplex of P (Theorem 5.1 and Lemma 10.2).Therefore, it is the convex hull of S n ∩ P erg . Because Γ has property (T), P erg is closedin P [GW97]. After passing to a subsequence we may assume that S n ∩ P erg converges tosome subset K ⊆ P erg as n → ∞ . But this implies S n converges to the convex hull of K ;and therefore S ∞ is the convex hull of K . So S ∞ is a subsimplex of P which implies that itis extreme by Lemma 10.2. 12 Groups with many extreme stable weak equivalenceclasses In [BG13], Brown and Guentner associate a C ∗ -algebra C ∗ D (Γ) to each algebraic ideal D in ℓ ∞ (Γ). We will be concerned with the case D = ℓ p (Γ) for 2 ≤ p < ∞ , and we write C ∗ ℓ p (Γ)for C ∗ ℓ p (Γ) (Γ), which is defined as follows. Definition 6 ([BG13]) . Let π be a unitary representation of Γ on a Hilbert space H π ,and let 2 ≤ p < ∞ . The representation π is said to be an ℓ p (Γ) -representation if thereexists a dense linear subspace H of H π such that for all ξ, η ∈ H the matrix coefficient π ξ,η : γ 7→ h π ( γ ) ξ, η i , belongs to ℓ p (Γ). The C ∗ -algebra C ∗ ℓ p (Γ) is defined as the completionof the group ring C [Γ] with respect to the C ∗ -norm k x k C ∗ ℓp := sup {k π ( x ) k : π is an ℓ p (Γ)-representation } , where k π ( x ) k denotes the operator norm of π ( x ).Since Γ is countable, and since the direct sum of ℓ p (Γ)-representations is an ℓ p (Γ)-representation, we can in fact find an ℓ p (Γ)-representation, denoted σ p Γ , on a separable Hilbert28pace H σ p Γ , such that k x k C ∗ ℓp = k σ p Γ ( x ) k for all x ∈ C [Γ]. Hence, C ∗ ℓ p (Γ) is isomorphic to the C ∗ -subalgebra of B ( H σ p Γ ) generated by σ p Γ (Γ). By [Dix77, Chapter 18] σ p Γ is uniquely definedup to weak equivalence of unitary representations, and, up to weak equivalence σ p Γ is theunique ℓ p (Γ)-representation which weakly contains all other ℓ p (Γ)-representations. If p ≤ q then k x k C ∗ ℓp ≤ k x k C ∗ ℓq for all x ∈ C [Γ], and the canonical quotient map from C ∗ ℓ q (Γ) onto C ∗ ℓ p (Γ) is an isomorphism if and only if σ p Γ and σ q Γ are weakly equivalent [Dix77]. The mainresult of this section is a direct consequence of the following striking result of Okayasu. Theorem 12.1 ([Oka14]) . Let F denote the free group on two generators and let ≤ p The representations κ a ( σ p Γ )0 and σ p Γ are weakly equivalent.Proof. Put σ = σ p Γ . By [KL16, Theorem E.19], κ a ( σ )0 contains σ and is isomorphic to asubrepresentation of L n ≥ ( σ ⊕ σ ) ⊗ n , where σ denotes the conjugate representation of σ . By[BG13], the representation L n ≥ ( σ ⊕ σ ) ⊗ n is an ℓ p (Γ)-representation, so (since it contains σ ) it is weakly equivalent to σ . Therefore, κ a ( σ )0 is weakly equivalent to σ as well.By [Dix77], every ℓ (Γ)-representation is a subrepresentation of a multiple of the leftregular representation of Γ. For concreteness, we will therefore take σ to be the left regularrepresentation of Γ. Also, for each 2 ≤ p < ∞ , since ℓ (Γ) ≤ ℓ p (Γ), we will assume (withoutloss of generality) that σ is a subrepresentation of σ p Γ . Then a ( σ ) is a Bernoulli shift action29f Γ, and for each 2 ≤ p < ∞ the action a ( σ p Γ ) factors onto a Bernoulli shift and hence isfree. Theorem 12.3. Let Γ be a group containing a subgroup isomorphic to F . Then the actions a ( σ p Γ ) , ≤ p < ∞ , are pairwise stably weakly inequivalent, and each is free, mixing andstrongly ergodic.Proof. We already observed that each of the actions a ( σ p Γ ) is free. Since Γ is non-amenable,the representation σ p Γ does not have almost invariant vectors [BG13]. Therefore, the repre-sentation κ a ( σ p Γ )0 , being weakly equivalent to σ p Γ , does not have almost invariant vectors. Thisimplies that a ( σ p Γ ) is strongly ergodic. 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