WEAK CONTAINMENT RIGIDITY FOR DISTAL ACTIONS. 1. Introduction

Transcription

1 WEAK CONTAINMENT RIGIDITY FOR DISTAL ACTIONS ADRIAN IOANA AND ROBIN TUCKER-DROB Abstract. We prove that f a measure dstal acto α of a coutable group Γ s weakly cotaed a strogly ergodc probablty measure preservg acto β of Γ, the α s a factor of β. I partcular, ths apples whe α s a compact acto. As a cosequece, we show that the weak equvalece class of ay strogly ergodc acto completely remembers the weak somorphsm class of the maxmal dstal factor arsg the Fursteberg-Zmmer Structure Theorem. 1. Itroducto The oto of weak cotamet for group actos was troduced by A. Kechrs [Ke10] as a aalogue of the oto of weak cotamet for utary represetatos. Let Γ α (X, µ) ad Γ β (Y, ν) be two probablty measure preservg (p.m.p.) actos of a coutable group Γ. The α s sad to be weakly cotaed β ( symbols, α β) f for ay fte set S Γ, fte measurable partto {A } =1 of X, ad ε > 0, we ca fd a measurable partto {B } =1 of Y such that for all γ S ad, j {1,..., } we have µ(γa A j ) ν(γb B j ) < ε. If α β ad β α, we say that α s weakly equvalet to β. We say that α s a factor of β f there exsts a measurable, measure preservg map θ : Y X such that θ(γy) = γθ(y), for all γ Γ ad almost every y Y. The map θ s called a factor map from β to α. If addto there s a coull set Y 0 Y such that θ s oe-to-oe o Y 0, the θ s called a somorphsm ad we say that α s somorphc to β. The actos α ad β are sad to be weakly somorphc f each s a factor of the other. As the termology suggests, f α s a factor of β, the α s weakly cotaed β. The ma goal of ths ote s to establsh a rgdty result whch provdes a geeral stace whe the coverse holds. Theorem 1.1. Let Γ be a coutable group, Γ α (X, µ) be a measure dstal p.m.p. acto, ad let Γ β (Y, ν) be a strogly ergodc p.m.p. acto. If α s weakly cotaed β, the α s a factor of β. I partcular, f a compact acto α s weakly cotaed a strogly ergodc acto β, the α s a factor of β. Weak cotamet ad weak equvalece have receved much atteto sce ther troducto. I [Ke10], A. Kechrs shows that cost vares mootocally wth weak cotamet ad [Ke12] Kechrs uses ths mootocty to obta a ew proof that free groups have fxed prce. Several other measurable combatoral parameters of actos are kow to respect weak cotamet ad hece are varats of weak equvalece; see [AE11, CK13, CKTD12]. A.I. was partally supported by NSF Grat DMS # , NSF Career Grat DMS # , ad a Sloa Foudato Fellowshp. R.T.D. was partally supported by NSF Grat DMS #

2 2 ADRIAN IOANA AND ROBIN TUCKER-DROB I [AW13], M. Abért ad B. Wess exhbt a remarkable at-rgdty pheomeo for weak cotamet by showg that every free p.m.p. acto of Γ weakly cotas the Beroull acto over a atomless base space. The Abért-Wess Theorem was exteded [TD15] ad used to show that every weak equvalece class cotas uclassfably may somorphsm classes of actos, thus rulg out the possblty of weak equvalece superrgdty. These at-rgdty results stad marked cotrast to the rgdty exhbted Theorem 1.1 ad Corollary 1.2 below, ad suggest that Theorem 1.1 ad Corollary 1.2 are lkely optmal. The frst weak cotamet rgdty results were obtaed by M. Abért ad G. Elek. They prove that f a fte acto α (.e. a acto o a fte probablty space) s weakly cotaed a strogly ergodc acto β, the α s a factor of β [AE10, Theorem 1]. From ths, they deduce that f two strogly ergodc profte actos are weakly equvalet, the they are somorphc [AE10, Theorem 2]. Recall that a p.m.p. acto s called profte f t s a verse lmt of fte actos. Sce ay profte acto s compact, Theorem 1.1 ad Corollary 1.2 below recover the results of [AE10]. Our results are ew for compact o-profte actos, ad partcular for traslato actos Γ (K, m K ) o coected compact groups K. Note that the approach of [AE10] reles o the fact that the case of profte actos Γ (X, µ), there are may Γ-varat measurable parttos of X. As such, t does ot apply to traslato actos o coected compact groups, sce these actos admt o o-trval varat measurable parttos. Istead, the proof of Theorem 1.1 reles o a ew, elemetary approach whch we brefly outle the case α s a compact acto. Assumg that α β, we fd a sequece θ : Y X of almost Γ-equvarat measurable maps. Sce β s strogly ergodc, the maps y d(θ m (y), θ (y)) must be asymptotcally costat, as m,. Whe coupled wth the compactess of the metrc space (X, d), ths forces a subsequece of {θ } to coverge almost everywhere. The lmt map θ : Y X s Γ-equvarat, ad hece realzes α as a factor of β. Recall that a p.m.p. acto Γ (Y, ν) s called strogly ergodc f ay sequece A Y of measurable sets satsfyg ν(γa A ) 0, for all γ Γ, s trval,.e. ν(a )(1 ν(a )) 0. A exteso (X, µ) (X 0, µ 0 ) of ergodc p.m.p. actos Γ (X, µ) ad Γ (X 0, µ 0 ) s called compact (or sometrc) f t s somorphc to a homogeeous skew-product exteso,.e., f there exst a compact group K, a closed subgroup L < K, ad a measurable cocycle w : Γ X 0 K, such that Γ (X, µ) s somorphc to Γ (X 0, µ 0 ) (K/L, m K/L ), where m K/L s the uque K-varat Borel probablty measure o K/L, the acto s gve by γ(x, kl) = (γx, w(γ, x)kl) (x X 0, kl K/L), ad where the map (X, µ) (X 0, µ 0 ) correspods to the projecto map (X 0, µ 0 ) (K/L, m K/L ) (X 0, µ 0 ). The ergodc acto Γ (X, µ) s called compact f the exteso (X, µ) ({ }, δ ), over a pot, s a compact exteso. Equvaletly, ths meas that the acto Γ (X, µ) s somorphc to a acto of the form Γ (K/L, m K/L ), where Γ acts by traslato o K/L va a homomorphsm Γ K wth dese mage. I partcular, ths case X ca be edowed wth a metrc d such that (X, d) s a compact metrc space o whch Γ acts sometrcally. The dstal tower assocated to a ergodc p.m.p. acto Γ α (X, µ) s the drected famly (Γ α ζ (X ζ, µ ζ )) ζ<ω1 of factors of α, satsfyg: (1.1) (1.2) (1.3) The acto α 0 s the trval acto o a pot mass; The acto α ζ+1 s the largest termedate compact exteso of α ζ wth α; For lmt ordals ζ, the acto α ζ s the verse lmt of (α ζ ) ζ <ζ. The least coutable ordal η for whch α η+1 = α η s called the order of the tower. The acto α s sad to be (measure) dstal f α = α ζ for some ζ < ω 1. The Fursteberg-Zmmer Structure

3 WEAK CONTAINMENT RIGIDITY 3 Theorem [Z76b,Fu77] states that every ergodc p.m.p. acto α of Γ has a uque maxmal dstal factor amely α η, where η s the order of the dstal tower assocated to α ad that α s relatvely weakly mxg over ths factor. I the rest of the troducto we preset several cosequeces of Theorem 1.1. The frst cosequece states that, for strogly ergodc actos, the weak somorphsm class of the maxmal dstal factor s a varat of weak equvalece. Corollary 1.2. Let Γ α (X, µ) ad Γ β (Y, ν) be p.m.p. actos of a coutable group Γ. Assume that β s strogly ergodc. If α s weakly equvalet to β, the the maxmal dstal factors of α ad β are weakly somorphc. Moreover, f β s compact ad strogly ergodc, the ay compact acto whch s weakly equvalet to β s fact somorphc to β. Remark. If two compact actos are weakly somorphc, the they are fact somorphc (see Lemma 2.7). Ths does ot exted to geeral dstal actos however; see [Le89] for a example of two dstal actos of Z whch are weakly somorphc but ot somorphc. Next, we relate Corollary 1.2 wth recet spectral gap results for traslato actos o coected compact groups. A p.m.p. acto Γ (X, µ) has spectral gap f the utary represetato Γ L 2 (X) C1 admts o almost varat vectors. If a acto has spectral gap, the t s strogly ergodc. Recet works of J. Bourga ad A. Gamburd [BG06, BG11], ad Y. Beost ad N. de Saxcé [BdS14] establshed spectral gap for a wde class of traslato actos. More precsely, t was show that f K s a smple coected compact Le group ad Γ < K s a coutable dese subgroup geerated by matrces wth algebrac etres, the the traslato acto Γ (K, m K ) has spectral gap (see [BdS14, Theorem 1.2]). Moreover, ths case, the left traslato acto Γ (K/L, m K/L ) has spectral gap, for ay closed subgroup L < K. Ths provdes a large famly of actos to whch Corollary 1.2 apples. I partcular, t allows us to costruct ew cocrete fte famles of weakly comparable traslato actos of free groups (cf. [AE10, Theorem 3]). To ths ed, let a, b be tegers such that 0 < a < b ad a b ± 1 2. Put c = b 2 a 2 ad let Γ be the subgroup of K = SO 3 (R) geerated by the followg rotatos: a b c b c A = a b b 0 ad B = a 0 b c b Deote by α a,b the assocated traslato acto Γ α (a,b) (K, m K ). For 2 +, deote by Γ the group geerated by {B k AB k 0 k 1}, ad by α (a,b) the restrcto of α (a,b) to Γ. Sce a b ± 1 2, [Sw94] shows that Γ s somorphc to F 2, ad therefore Γ m s somorphc to F. Corollary 1.3. Let (a, b), (a, b ) be two pars of tegers as above, ad 2 +. If a b a b, the α (a,b) α (a,b ) ad α (a,b) α (a,b ). L. Bowe recetly proved that f β s ay essetally free p.m.p. acto of a free group Γ = F, for some 2 +, the ts orbt equvalece class s weakly dese the space of all p.m.p. actos of Γ (see [Bo13, Theorem 1.1]). I other words, for ay p.m.p. acto α of Γ, there exsts a sequece {β } of actos of Γ whch are orbt equvalet to β ad coverge to α, the weak topology. As a cosequece, α s weakly cotaed the fte drect product acto =1 β. I vew of ths result, t s atural to woder whether the sequece {β } ca be take costat, that s whether α s weakly cotaed some acto whch s orbt equvalet to β. By combg c b a b.

4 4 ADRIAN IOANA AND ROBIN TUCKER-DROB Theorem 1.1 wth a result of I. Chfa, S. Popa ad O. Szemore [CPS11], we are able to show that ths s ot the case. More geerally, we have: Corollary 1.4. Let Γ α (X, µ) be a essetally free ergodc compact p.m.p. acto of a coutable o-ameable group Γ. Let Λ β (Y, ν) = (Z, ρ) Λ be a Beroull acto of a coutable group Λ. The there does ot exst a p.m.p. acto Γ σ (Y, ν) such that α s weakly cotaed σ, ad σ(γ)(y) β(λ)(y), for almost every y Y. Ackowledgemet. We are very grateful to Bejy Wess for potg a error the frst verso of the paper. 2. Proofs I ths secto we prove the results stated the troducto. We start wth some termology related to weak cotamet. Defto 2.1. Let (Y, ν) be a probablty space ad let X be a measurable space. () A sequece θ : Y X, N, of measurable maps s sad to coverge weakly to the measurable map θ : Y X, f ν(θ 1 (A) θ 1 (A)) 0 for all measurable subsets A X. () Let Γ α (Y, ν) ad Γ β X be measurable actos of Γ wth α probablty measure preservg. A sequece θ : Y X, N, of measurable maps s sad be asymptotcally equvarat for the actos f ν(θ 1 (γ 1 A) γ 1 θ 1 (A)) 0 for all γ Γ ad measurable subsets A X. We ow have the followg useful characterzato of weak cotamet. Lemma 2.2. [Ke10] Let Γ α (X, µ) ad Γ β (Y, ν) be p.m.p. actos. The α β f ad oly f there s a sequece θ : (Y, ν) (X, µ), N, of measure preservg maps whch are asymptotcally equvarat. Proof. See the proof of [Ke10, Proposto 10.1]. Lemma 2.3. Let (Y, ν) be a probablty space, let (X, d) be a Polsh metrc space wth d 1, ad let µ a Borel probablty measure o X. (1) Let (ψ ) N ad (ϕ ) N be two sequeces of measure preservg maps from (Y, ν) to (X, µ). The Y d(ψ (y), ϕ (y)) dν 0 f ad oly f ν(ψ 1 (A) ϕ 1 (A)) 0 for all measurable subsets A X. (2) Let Γ α (X, µ) ad Γ β (Y, ν) be p.m.p. actos ad let θ : (Y, ν) (X, µ), N, be a sequece of measure preservg maps. The sequece (θ ) N s asymptotcally equvarat f ad oly f Y d(θ (γy), γθ (y)) dν 0 for each γ Γ. Moreover, f (θ ) N s asymptotcally equvarat the we may fd a subsequece (θ k ) k N such that d(θ k (γy), γθ k (y)) 0, for all γ Γ ad almost every y Y. Proof. (1): Assume frst that ν(ψ 1 (A) ϕ 1 (A)) 0 for all measurable subsets A X. Fx ε > 0. Let {A } k =1 be a collecto of parwse dsjot Borel subsets of X such that each A has dameter at most ε ad ν( k =1 A ) > 1 ε. Put A 0 = X \ k =1 A. If j, the {y Y ψ (y) A, ϕ (y) A j } ψ 1 (A ) \ ϕ 1 (A ).

5 WEAK CONTAINMENT RIGIDITY 5 Hece ν({y Y ψ (y) A, ϕ (y) A j }) 0. Thus, we coclude that ν ( k {y Y ψ (y) A, ϕ (y) A } ) 1. =0 Sce ν(ψ 1 (A 0 )) = µ(a 0 ) ε, we get lm f ν ( k =1 {y Y ψ (y) A, ϕ (y) A } ) 1 ε. O the other had, f ψ (y), ϕ (y) A for some 1 k, the d(ψ (y), ϕ (y)) ε. Ths proves that lm f ν({y Y d(ψ (y), ϕ (y)) ε}) 1 ε, for every ε > 0, ad hece Y d(ψ (y), ϕ (y)) dν 0. Coversely, assume that Y d(ψ (y), ϕ (y)) dν 0. Fx A X measurable, ad ε > 0. Sce (X, d) s Polsh, the measure µ s tght ad regular, so we may fd L A U wth L compact, U ope, ad µ(u \ L) < ε. Lettg r = d(l, X \ U) > 0, we have ψ 1 (L) \ ϕ 1 (U) {y Y d(ψ (y), ϕ (y)) r}, ad hece ν(ψ 1 (L) \ ϕ 1 (U)) 0. Therefore, sce ψ ad ϕ are measure preservg, the cotamet ψ 1 (A) \ ϕ 1 (A) [ψ 1 (L) \ ϕ 1 (U)] [ψ 1 (A \ L)] [ϕ 1 (U \ A)] mples that lm sup ν(ψ 1 (A) \ ϕ 1 (A)) µ(u \ L) < ε. Sce ε > 0 was arbtrary we coclude that ν(ψ 1 (A) \ ϕ 1 (A)) 0, ad hece ν(ψ 1 (A) ϕ 1 (A)) 0. The frst part of (2) s mmedate from (1), ad the fal statemet s clear. The proof of Theorem 1.1 reles o the followg result. Lemma 2.4. Let Γ β (Y, ν) be a strogly ergodc p.m.p. acto. Let (X, d) be a compact metrc space, K the group of sometres of X, ad w : Γ Y K a measurable cocycle. Assume that there exsts a sequece θ : Y X, N, of measurable maps such that Y d(θ (γy), w(γ, y)θ (y)) dν(y) 0, for all γ Γ. The there exsts a measurable map θ : Y X such that θ(γy) = w(γ, y)θ(y), for all γ Γ ad almost every y Y, ad there s a subsequece (θ k ) k N whch coverges to θ both weakly ad potwse. Proof. We may clearly assume d 1. Gve two measurable maps θ, θ : Y X, we defe d(θ, θ ) = d(θ(y), θ (y)) dν(y). Y Clam. Let θ : Y X be a sequece of measurable maps ad assume that for each γ Γ we have Y d(θ (γy), w(γ, y)θ (y)) dν(y) 0. The for every ε > 0, there exsts 1 such that the set {m d(θ m, θ ) < ε} s fte. Assumg the clam, let us derve the cocluso. By usg the clam we ca ductvely costruct a subsequece {θ k } of {θ } such that d(θ k+1, θ k ) < 1, for ay k 1. Ths mples that 2 k d(θ k+1 (y), θ k (y)) dν(y) = d(θ k+1, θ k ) < 1. Y k=1 Therefore, the sequece {θ k (y)} X s Cauchy ad thus coverget, for almost every y Y. It s clear that the map θ : Y X defed as θ(y) := lm k θ k (y) satsfes the cocluso. Proof of the clam. Suppose that the clam s false. Thus, there exsts ε 0 > 0 such that for all 1, the set {m d(θ m, θ ) < ε 0 } s fte. The we ca ductvely fd a subsequece {θ k } of {θ } such that d(θ k, θ l ) ε 0, for all l > k 1. k=1

6 6 ADRIAN IOANA AND ROBIN TUCKER-DROB Sce X s compact, t ca be covered by ftely may, say p 1, balls of radus ε 0 4. Therefore, f x 0, x 1,..., x p are p + 1 pots X, the d(x, x j ) ε 0 2, for some 0 < j p. Next, for m, 1, defe f m, : Y [0, 1] by lettg f m, (y) = d(θ m (y), θ (y)). The for all γ Γ ad y Y we have that f m, (γy) f m, (y) = d(θ m (γy), θ (γy)) d(w(γ, y)θ m (y), w(γ, y)θ (y)) d(θ m (γy), w(γ, y)θ m (y)) + d(θ (γy), w(γ, y)θ (y)). From ths t follows that Y f m,(γy) f m, (y) dν(y) 0, for all γ Γ, as m,. Sce f m, 1, for all m,, the strog ergodcty assumpto mples that f m, Y f m, 1 0, as m,. I other words, d(θ m (y), θ (y)) d(θ m, θ ) dν(y) 0, as m,. Y Recallg that d(θ k, θ l ) ε 0, for all l > k 1, t follows that Ths further mples that ν({y Y d(θ k (y), θ l (y)) > ε 0 }) 1, as k, l. 2 ν({y Y d(θ k+ (y), θ k+j (y) > ε 0, for all 0 < j p}) 1, as k. 2 I partcular, f k s large eough, the we ca fd y Y such that d(θ k+ (y), θ k+j (y)) > ε 0 2, for all 0 < j p. Ths cotradcts the choce of p, ad fshes the proof of the clam. The followg cosequece of Lemma 2.4 mght be of depedet terest. Corollary 2.5. Let Γ (Y, ν) be a strogly ergodc p.m.p. acto. Let K a compact metrzable group edowed wth a left-rght varat compatble metrc d. Let w : Γ Y K be a measurable cocycle, ad L < K be a closed subgroup. Assume that there exsts a sequece of measurable maps φ : Y K such that d(φ (γy) 1 w(γ, y)φ (y), L) 0, for all γ Γ ad almost every y Y. The there exsts a measurable map φ : Y K such that φ(γy) 1 w(γ, y)φ(y) L, for all γ Γ ad almost every y Y. Ths result geeralzes [Sc80, Proposto 2.3] whch dealt wth the case L = {e} ad K = T. It also exteds [Io13, Lemma J], where t was otced that the proof of [Sc80] ca be adapted to more geerally treat the case whe L = {e} ad K s ay compact metrzable group. Proof. Edow X = K/L wth the metrc d(xl, yl) := d(x, yl) = f{d(x, yl) l L}. The (X, d) s a compact metrc space ad the left multplcato acto K X s sometrc. Defe θ : Y X by lettg θ (y) = φ (y)k. The for all γ Γ ad almost every y Y we have lm d(θ (γy), w(γ, y)θ (y)) = lm d(φ (γy) 1 w(γ, y)φ (y), L) = 0. By Lemma 2.4 we deduce that there exsts a measurable map θ : Y X such that θ(γy) = w(γ, y)θ(y), for all g Γ ad almost every y Y. If φ : Y K s a Borel map such that θ(y) = φ(y)k, for all y Y, the the cocluso follows. Lemma 2.6. Let Γ β (Y, ν) be a strogly ergodc p.m.p. acto ad let Γ α (X, µ) be a dstal p.m.p. acto. Assume that α β, as wtessed by the asymptotcally equvarat sequece θ : (Y, ν) (X, µ), N, of measure preservg maps. The there exsts a factor map θ : (Y, ν) (X, µ) from β to α, alog wth a subsequece (θ k ) whch coverges weakly to θ.

7 WEAK CONTAINMENT RIGIDITY 7 Proof. Let (Γ α ζ (X ζ, µ ζ )) ζ<ω1 be the dstal tower assocated to α, as (1.1)-(1.3), ad let η < ω 1 be the order of the tower. Sce α s dstal we have α = α η. We prove the lemma by trasfte ducto o η. If η = 0 the the statemet s obvous, so assume that η > 0. Case 1: η = η s a successor ordal. I ths case the exteso (X, µ) (X η0, µ η0 ) s compact, so we may assume wthout loss of geeralty that (X, µ) = (X η0, µ η0 ) (K/L, m K/L ), ad that the acto α s of the form γ(x, kl) = (γx, w(γ, x)kl) for some measurable cocycle w : Γ X η0 K. For each we may wrte θ (y) = (θ 0 (y), θ 1 (y)), where θ 0 : Y X η0 ad θ 1 : Y K/L are the compostos of θ wth the left ad rght projectos respectvely. Applyg the ducto hypothess to the acto α η0 ad the sequece (θ 0 ), we obta a subsequece (θ 0 ) ad a factor map θ 0 : (Y, ν) (X η0, µ η0 ) such that θ 0 coverges weakly to θ 0. Defe θ : (Y, ν) (X, µ) by θ (y) = (θ 0 (y), θ(y)), 1 so that the subsequece θ s asymptotcally equvarat. Fx a compatble Polsh metrc d 0 1 o (X η0, µ η0 ), let d K/L 1 be a compatble K-varat metrc o K/L, ad let d be the compatble Polsh metrc o X gve d((x, kl), (x, k L)) = 1 2 (d 0(x, x ) + d K/L (kl, k L)). By Lemma 2.3, for each γ Γ we have lm Y d( θ (γy), γ θ (y)) dν(y) = 0. Sce θ 0 s Γ-equvarat ths meas that for each γ Γ we have d K/L (θ 1 (γy), w(γ, θ 0 (y))θ 1 (y)) dν(y) = 0. lm Y Applyg Lemma 2.4, we obta a measurable map θ 1 : Y K/L wth θ 1 (γy) = w(γ, θ 0 (y))θ 1 (y), alog wth a subsequece (θ 1 ) of (θ 1 ) wth Y d K/L(θ 1 θ (y), 1 (y)) dν 0. Therefore, the map θ : (Y, ν) (X, µ) defed by θ(y) = (θ 0 (y), θ 1 (y)) s a factor map from β to α, ad by Lemma 2.3, the subsequece (θ ) coverges weakly to θ. Case 2: η s a lmt ordal. Fx a sequece (ζ j ) j N of ordals whch strctly crease to η. For each j < k N let ϕ j,k : (X ζk, µ ζk ) (X ζj, µ ζj ) deote the factor map from α ζk to α ζj, ad let ϕ j : (X, µ) (X ζj, µ ζj ) deote the factor map from α = α η to α ζj. Applyg the ducto hypothess to the acto α ζ0 ad the asymptotcally equvarat sequece ϕ 0 θ : (Y, ν) (X ζ0, µ ζ0 ), we obta a factor map θ 0 : (Y, ν) (X ζ0, µ ζ0 ) ad a subsequece ( 0 ) N such that ϕ 0 θ 0 coverges weakly to θ 0. Havg defed the factor map θ j : (Y, ν) (X ζj, µ ζj ) ad subsequece ( j ) N, we apply the ducto hypothess to acto α ηj+1 ad the sequece (ϕ j+1 θ j) N to obta a factor map θ j+1 : (Y, ν) (X ζj+1, µ ζj+1 ) ad a subsequece ( j+1 ) N of ( j ) N wth ϕ j+1 θ j+1 covergg weakly to θ j+1. Observe that ϕ j,k θ k = θ j for all j < k N. Sce α = lm α j ζj, ths mples that there s a uque factor map θ = lm θ j from β j to α such that ϕ j θ = θ j for all j N. Morover, f we defe the subsequece ( ) N by =, the for each j N, the sequece (ϕ j θ ) N coverges weakly to θ j, ad hece the sequece (θ ) N coverges weakly to θ. Proof of Theorem 1.1. Ths s mmedate from Lemma 2.6. Lemma 2.7. Let Γ α (X, µ) ad Γ β (Y, ν) be ergodc compact p.m.p. actos of Γ. If α are β are weakly somorphc, the they are somorphc. Proof. Let ψ 0 : (X, µ) (Y, ν) ad ψ 1 : (Y, ν) (X, µ) be factor maps from α to β ad from β to α respectvely. It suffces to show that the map θ = ψ 1 ψ 0, factorg α oto tself, s a somorphsm. Sce α s a ergodc compact acto, we may assume that (X, µ) = (K/L, m K/L ), where K s a compact metrzable group, L < K s a closed subgroup, ad that Γ acts by traslato o K/L va a homomorphsm Γ K wth dese mage ( what follows we wll detfy Γ wth ts mage K). I order to show that θ s jectve o a coull subset of K/L,

8 8 ADRIAN IOANA AND ROBIN TUCKER-DROB t s eough to show that the sometrc lear embeddg T θ : L 2 (K/L) L 2 (K/L), ξ ξ θ, whch θ duces o L 2 (K/L), s surjectve. The traslato acto K K/L gves rse to a utary represetato λ K/L, of K o H = L 2 (K/L), whch we may vew as a subrepresetato of the left regular represetato of K of L 2 (K) va the cluso L 2 (K/L) L 2 (K) assocated to the atural projecto K K/L. The represetato λ K/L may be expressed as a drect sum λ K/L = π K λπ K/L, where K s a collecto of represetatves for somorphsm classes of rreducble utary represetatos of K, ad where for each π K, the represetato λ π K/L s the restrcto of λ K/L to the closed lear spa H π, of all subspaces of H o whch λ K/L s somorphc to π. For each π K we may further wrte H π as a drect sum of rreducble subspaces H π = < π H π,, where for each < π the represetatos λ π, K/L := λ K/L H π, s somorphc to π. Sce λ K/L s a subrepresetato of the left regular represetato of K, by the Peter-Weyl Theorem, each π s fte ad therefore the subspaces H π are all fte dmesoal. Sce Γ s dese K, the operator T θ tertwes λ K/L wth tself, so by Schur s Lemma we see that T θ tertwes each λ π K/L wth tself,.e., T θ (H π ) H π for all π K. Sce T θ s jectve ad each H π s fte dmesoal t follows that T θ (H π ) = H π for all π K ad hece T θ s surjectve o L 2 (K/L), as was to be show. Proof of Corollary 1.2. Assume that α s weakly equvalet to β. Note that ths mples that α s strogly ergodc, sce β s strogly ergodc. Let α ad β be the maxmal dstal factors of α ad β respectvely. As α s a factor of α, ad α β, we have that α β. Therefore, α s a factor of β by Theorem 1.1, ad hece α (beg a dstal factor of β) s a factor of β. Lkewse, reversg the roles of α ad β, we obta that β s a factor of α. Thus, α ad β are weakly somorphc. The fal statemet of Corollary 1.2 ow follows from Lemma 2.7. Proof of Corollary 1.3. Assume that α (a,b) α (a,b ). Deote by A, B ad A, B the matrces costructed from the pars (a, b) ad (a, b ). Sce A, B have algebrac etres, α (a,b ) has spectral gap (see [BdS14, Theorem 1.2]). Corollary 1.2 gves that α (a,b) s a factor of α (a,b ). Let θ : K K be a Γ-equvarat measurable map. If t K, the K x θ(x) 1 θ(xt) K s a Γ-varat map. Thus, there s δ(t) K such that θ(x) 1 θ(xt) = δ(t), for almost every x K. It follows that δ : K K s a cotuous homomorphsm whose restrcto to Γ s the detty. I other words, δ satsfes δ(a) = A ad δ(b) = B. Sce K s a smple group, δ s oe-to-oe. Sce K s ot somorphc to ay of ts proper closed subgroups, δ s also oto. Thus, sce K has o outer automorphsms, we ca fd g K such that δ(x) = gxg 1, for all x K. I partcular, A, A must have the same trace, hece a b = a b. Smlarly, f α (a,b) α (a,b ), for some 2, t follows that a b = a b. Ths completes the proof. Proof of Corollary 1.4. Assume by cotradcto that Γ σ (Y, ν) s a p.m.p. acto such that α σ ad σ(γ)(y) β(λ)(y), for almost every y Y. Idetfy α wth a left traslato acto Γ (K/L, m K/L ) assocated to a dese embeddg of Γ to a compact group K. Let d be a compatble metrc o X = K/L. By Lemma 2.3 we ca fd a sequece θ : Y X of measurable maps such that d(θ (γy), γθ (y)) 0, for all γ Γ ad almost every y Y. Deote by R σ ad R β the equvalece relatos assocated to σ ad β, so that R σ R β. Sce α s essetally free ad α σ, σ s essetally free. Sce Γ s o-ameable, the restrcto of R σ to ay o-eglgble set Y 0 Y s ot hyperfte. By [CI08, Theorem 1] we deduce the exstece of a σ(γ)-varat o-eglgble measurable set Y 1 Y such that the restrcto σ 1 of σ to Y 1 s strogly ergodc. By applyg Lemma 2.4 to the restrctos of θ to Y 1 we coclude

9 WEAK CONTAINMENT RIGIDITY 9 that α s a factor of σ 1. Let θ : Y 1 X be a measurable, measure preservg map such that θ(γy) = γθ(y), for every γ Γ ad almost every y Y 1. We wll reach a cotradcto by applyg [CPS11, Theorem 6.2]. To ths ed, we deote by Θ : L (X) α Γ L (Y 1 ) σ1 Γ the -homomorphsm gve by Θ(f) = f θ ad Θ(u γ ) = u γ, for all f L (X) ad γ Γ. We vew N := L (Y ) σ Γ as a subalgebra of M := L (Y ) β Λ. Let p = 1 Y1 N, P = Θ(L (X)) pmp ad G = {au γ p a U(L (Y )), γ Γ}. The G s a group of utares pmp whch ormalze P ad satsfes G = Np. Next, deote by M = L(Z Λ) the vo Neuma algebra of the wreath product group Z Λ. Recall that (Y, ν) = (Z, ρ) Λ ad cosder a fxed embeddg of L (Z) to L(Z). From ths we get a Λ-equvarat embeddg of L (Y ) = L (Z) Λ to L( λ Λ Z) = L(Z) Λ, ad thus a embeddg M M. It s easy to that (L (Y )p) p Mp = L( λ Λ Z)p. Thus, f we deote Q = G p Mp, the Q L( λ Λ Z)p. Sce Q commutes wth Np ad N has o ameable drect summad, [CI08, Theorem 2] mples that Q s completely atomc. Let q Q be a o-zero projecto such that Qq = Cq. The (Gq) q Mq = Cq. Morever, sce α s a compact acto, the cojugato acto G P s compact. Hece, the cojugato acto Gq P q s compact ad thus weakly compact (see [CPS11, Defto 6.1]). By applyg [CPS11, Theorem 6.2] we get that ether (Gq) = Nq has a ameable drect summad or P L(Λ) M (see [CPS11, Defto 2.1]). Sce N has o ameable drect summad ad P L( λ Λ Z), ether of these codtos hold true, whch gves the desred cotradcto. Refereces [AE10] M. Abért, G. Elek: Dyamcal propertes of profte actos, Erg. Th. Dyam. Sys. 32 (2012), [AE11] M. Abért, G. Elek: The space of actos, partto metrc ad combatoral rgdty, arxv preprt arxv: (2011). [AW13] M. Abért, B. Wess: Beroull actos are weakly cotaed ay free acto, Erg. Th. Dyam. Sys. 33 (2013), [BdS14] Y. Beost, N. de Saxcè: A spectral gap theorem smple Le groups, preprt arxv: [BG06] J. Bourga, A. Gamburd: O the spectral gap for ftely-geerated subgroups of SU(2), Ivet. Math. 171 (2008), [BG11] J. Bourga, A. Gamburd: A spectral gap theorem SU(d), J. Eur. Math. Soc. (JEMS) 14 (2012), [Bo13] L. Bowe: Weak desty of orbt equvalece classes of free group actos, to appear Groups Geom. Dy., preprt arxv: [CI08] I. Chfa, A. Ioaa: Ergodc subequvalece relatos duced by a Beroull acto, Geom. Fuct. Aal. 20 (2010), [CK13] C. Coley, A. Kechrs: Measurable chromatc ad depedece umbers for ergodc graphs ad group actos, Groups Geom. Dy. 7 (2013), [CKTD12] C. Coley, A. Kechrs, R. Tucker-Drob: Ultraproducts of measure preservg actos ad graph combatorcs, Erg. Th. Dyam. Sys. 33 (2012), [CPS11] I. Chfa, S. Popa, O. Szemore: Some OE- ad W -rgdty results for actos by wreath product groups, J. Fuct. Aal. 263 (2012), [Fu77] H. Fursteberg: Ergodc behavor of dagoal measures ad a theorem of Szemeréd o arthmetc progressos. J. Aal. Math. 31 (1977), [Gl03] E. Glaser: Ergodc theory va jogs, Mathematcal Surveys ad Moographs, 101. Amerca Mathematcal Socety, Provdece, RI [Io13] A. Ioaa: Orbt equvalece ad Borel reducblty rgdty for profte actos wth spectral gap, preprt arxv: [Ke10] A. Kechrs: Global aspects of ergodc group actos, Mathematcal Surveys ad Moographs, 160. Amerca Mathematcal Socety, Provdece, RI, [Ke12] A. Kechrs: Weak cotamet the space of actos of a free group, Israel J. of Math. 189 (2012),

10 10 ADRIAN IOANA AND ROBIN TUCKER-DROB [Le89] M. Lemańczyk: O the weak somorphsm of strctly ergodc homeomorphsms, Moatshefte für Mathematk 108 (1989), [Sc80] K. Schmdt: Asymptotcally varat sequeces ad a acto of SL(2, Z) o the 2-sphere, Israel J. Math. 37 (1980), o. 3, [Sw94] S. Śwerczkowsk: A class of free rotato groups, Idag. Math., Volume 5, Issue 2, 1994, [TD15] R. Tucker-Drob: Weak equvalece ad o-classfablty of measure preservg actos, Erg. Th. Dyam. Sys. 35 (2015), [Z76a] R. Zmmer: Extesos of ergodc group actos, Ill. J. of Math., 20 (1976), [Z76b] R. Zmmer: Ergodc actos wth geeralzed dscrete spectrum, Ill. J. of Math., 20 (1976), A.I. Mathematcs Departmet; Uversty of Calfora, Sa Dego, CA (Uted States) R.T.D. Departmet of Mathematcs; Rutgers Uversty, Pscataway, NJ , U.S.A. E-mal address: aoaa@ucsd.edu, rtuckerd@math.rutgers.edu