Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
aa r X i v : . [ m a t h . D S ] J un STRONG ERGODICITY, PROPERTY (T), AND ORBIT EQUIVALENCERIGIDITY FOR TRANSLATION ACTIONS
ADRIAN IOANA
Abstract.
We study equivalence relations that arise from translation actions Γ y G which areassociated to dense embeddings Γ < G of countable groups into second countable locally compactgroups. Assuming that G is simply connected and the action Γ y G is strongly ergodic, we provethat Γ y G is orbit equivalent to another such translation action Λ y H if and only if there existsan isomorphism δ : G → H such that δ (Γ) = Λ. If G is moreover a real algebraic group, then weestablish analogous rigidity results for the translation actions of Γ on homogeneous spaces of theform G/ Σ, where Σ < G is either a discrete or an algebraic subgroup. We also prove that if G issimply connected and the action Γ y G has property (T), then any cocycle w : Γ × G → Λ withvalues into a countable group Λ is cohomologous to a homomorphism δ : Γ → Λ. As a consequence,we deduce that the action Γ y G is orbit equivalent superrigid: any free nonsingular action Λ y Y which is orbit equivalent to Γ y G , is necessarily conjugate to an induction of Γ y G . Introduction and statement of main results
In the last 15 years there have been many exciting developments in the study of equivalence relationsarising from nonsingular actions of countable groups (see the surveys [Po07, Fu09, Ga10]). Thegoal of the present work is to investigate the equivalence relations that are associated to denseembeddings of countable groups into locally compact groups. More precisely, consider a locallycompact second countable (abbreviated, l.c.s.c. ) group G endowed with a left Haar measure m G ,and a countable dense subgroup Γ < G . Then the left translation action Γ y G is measurepreserving and ergodic. We denote by R (Γ y G ) the orbit equivalence relation on G of belongingto the same Γ-orbit: x ∼ y ⇐⇒ Γ x = Γ y . This paper is motivated by the following question: towhat extent does R (Γ y G ) remember the original inclusion Γ < G it was constructed from?In the case G is compact , this question has been investigated recently by the author [Io08,Io13] andby A. Furman [Fu09]. Firstly, it was shown that if Γ has property (T), then the action Γ y ( G, m G )is orbit equivalent superrigid : any probability measure preserving action Λ y ( Y, ν ) which is orbitequivalent to Γ y ( G, m G ) must be virtually conjugate to it. This result was obtained in [Io08],when G is a profinite group, and in [Fu09], for general compact groups G . Secondly, assume that G is a profinite group and that Γ y ( G, m G ) has spectral gap. Then the action Γ y ( G, m G ) was veryrecently shown to satisfy the following rigidity statement: any translation action Λ y ( H, m H ) ona profinite group H that is orbit equivalent to Γ y ( G, m G ) must be virtually conjugate to it [Io13].On the other hand, in the case G is locally compact but non-compact , little is known about howproperties of the inclusion Γ < G reflect on the structure of R (Γ y G ). In fact, even basicquestions, that can be easily answered in the compact case, are extremely challenging in the non-compact case. This is best illustrated by the question of characterizing when is the equivalencerelation R (Γ y G ) amenable (or, equivalently by [CFW81], hyperfinite). If G is compact, then R (Γ y G ) is amenable if and only if Γ is amenable. Only recently, by combining the structure The author was partially supported by NSF Grant DMS theory of locally compact groups [MZ55] with their topological Tits alternative, E. Breuillard andT. Gelander were able to answer this question for arbitrary locally compact groups G . They showedthat R (Γ y G ) is amenable if and only if there exists an open subgroup G < G such that Γ ∩ G is amenable [BG04]. Note that in the case G is a connected Lie group, this result was establishedearlier by R. Zimmer in [Zi87] (see also [CG85], for a proof in the case G = SL ( R ), and [BG02],for an alternative proof of the general case).In this paper, we investigate R (Γ y G ) for general locally compact groups G , and obtain severalrigidity results. Roughly speaking, these results provide instances when the inclusion Γ < G canbe partially, or even entirely, recovered from R (Γ y G ). More precisely, if the action Γ y ( G, m G )verifies certain conditions that strengthen non-amenability (strong ergodicity/property (T)), weprove that the equivalence relation R (Γ y G ) satisfies rigidity/superrigidity statements analogousto the ones obtained in [Io08, Fu09, Io13] in the case when G compact (see Theorems A, B and C).Our method of proof relies on a result for untwisting cocycles w : Γ × G → Λ with values intocountable groups Λ (see Theorem 3.1). As such, we are able to more generally study equivalencerelations R (Γ y G/ Σ) which arise from translation actions on homogeneous spaces G/ Σ, whereΣ < G is a closed subgroup (see Theorems E and F).Before stating our main results in detail, we first review some terminology, starting with the notionsof nonsingular actions and orbit equivalence.1.1.
Nonsingular actions and orbit equivalence.
Let G be a l.c.s.c. group and X be a standardBorel space together with a Borel action G y X . A σ -finite Borel measure µ on X is quasi-invariant under the G -action if µ ( A ) = 0 = ⇒ µ ( gA ) = 0, for every measurable set A ⊂ X and all g ∈ G .The measure µ is called invariant if µ ( gA ) = µ ( A ), for every measurable set A ⊂ X and all g ∈ G .If µ is quasi-invariant, then the action G y ( X, µ ) is called nonsingular . If µ is invariant, thenthe action G y ( X, µ ) is called measure preserving . In this case, if µ is a probability measure, wesay that the action is probability measure preserving (abbreviated, p.m.p. ). A nonsingular action G y ( X, µ ) is called free if the stabilizer in G of almost every point is trivial, and ergodic if any G -invariant measurable set A ⊂ X is either null or conull, i.e. µ ( A )(1 − µ ( A )) = 0. Example 1.1.
Let G be a l.c.s.c. group and Σ < G a closed subgroup. Then there exists aBorel measure µ on G/ Σ which is quasi-invariant under the action G y G/ Σ. Moreover, any twosuch measures µ , µ ′ are equivalent, i.e. they have the same null sets. From now on, we fix aquasi-invariant measure on G/ Σ, which we denote by m G/ Σ . Whenever G/ Σ admits a G -invariantmeasure, we choose m G/ Σ to be G -invariant. In particular, m G denotes a left Haar measure of G .In this paper, we study left translation actions of the form Γ y ( G/ Σ , m G/ Σ ), where Γ < G is acountable subgroup. If Γ < G is dense (as we will typically assume), then this action is ergodic. Definition 1.2.
Two nonsingular actions Γ y ( X, µ ), Λ y ( Y, ν ) of two countable groups Γ , Λ onstandard measure spaces (
X, µ ) , ( Y, ν ) are said to be orbit equivalent (OE) if there is a nonsingularisomorphism θ : X → Y such that θ (Γ x ) = Λ θ ( x ), for almost every x ∈ X . The actions are called stably orbit equivalent (SOE) if there are non-negligible measurable sets A ⊂ X , B ⊂ Y and anonsingular isomorphism θ : A → B such that θ (Γ x ∩ A ) = Λ θ ( x ) ∩ B , for almost every x ∈ A .Finally, the actions are called conjugate if there exist a nonsingular isomorphism θ : X → Y and agroup isomorphism δ : Γ → Λ such that θ ( gx ) = δ ( g ) θ ( x ), for all g ∈ Γ and almost every x ∈ X .1.2. Strong ergodicity and property (T).
In order to formulate our main results, we also needto recall the notion of strong ergodicity and property (T) for nonsingular actions.
Definition 1.3. [CW81,Sc81] Let Γ be a countable group and Γ y ( X, µ ) be a nonsingular ergodicaction on a standard probability space (
X, µ ). A sequence { A n } of measurable subsets of X is said RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 3 to be asymptotically invariant ( a.i. ) if lim n →∞ µ ( gA n ∆ A n ) = 0, for all g ∈ Γ. The action Γ y ( X, µ )is called strongly ergodic if any sequence { A n } of a.i. sets is trivial, i.e. lim n →∞ µ ( A n )(1 − µ ( A n )) = 0.Strong ergodicity only depends on the measure class of µ . This observation allows to extend thenotion of strong ergodicity to actions on infinite measure spaces. A nonsingular action Γ y ( X, µ )on a standard (possibly infinite) measure space (
X, µ ) is said to be strongly ergodic if it is stronglyergodic with respect to a probability measure µ which is equivalent to µ . Definition 1.4. [Zi81] Let Γ be a countable group. A nonsingular action Γ y ( X, µ ) is said tohave property (T) if any cocycle c : Γ × X → U ( H ) into the unitary group of a Hilbert space H which admits a sequence of almost invariant unit vectors necessarily has an invariant unit vector.For the notions of almost invariant and invariant unit vectors, see Section 2.4. For now, note thata p.m.p. action Γ y ( X, µ ) has property (T) if and only if the acting group Γ has property (T) ofKazhdan. This fact is however no longer true if the action is not p.m.p.1.3.
Orbit equivalence rigidity for translation actions.
We are now ready to state our mainresults. Our first result is a “locally compact analogue” of [Io13, Theorem A and Corollary 6.3]which established similar statements in the case G and H are profinite or connected compact groups. Theorem A (OE rigidity, I) . Let G be a simply connected l.c.s.c. group and Γ < G a countabledense subgroup. Assume that the translation action Γ y ( G, m G ) is strongly ergodic. Let H be asimply connected l.c.s.c. group and Λ < H a countable subgroup.Then the actions Γ y ( G, m G ) and Λ y ( H, m H ) are SOE if and only if there exists a topologicalisomorphism δ : G → H such that δ (Γ) = Λ . We continue with two cocycle and orbit equivalence superrigidity results for translation actions onlocally compact groups. These are analogous to the results proved in [Io08, Fu09] in the case oftranslation actions of property (T) groups Γ on compact groups G . When G is locally compactbut not compact, the assumption that Γ has property (T) needs to be replaced with the strongerassumption that the (infinite measure preserving) action Γ y ( G, m G ) has property (T). Thenecessity of imposing a property (T) condition on the action was inspired by an analogous situationfor weakly mixing s-malleable measure preserving actions Γ y ( X, µ ). These actions were originallyshown to be U fin -cocycle superrigid whenever Γ has property (T) and µ ( X ) < + ∞ [Po05]. Lateron, cocycle superrigidity for Γ y ( X, µ ) was proved for possibly infinite measure spaces (
X, µ ),whenever the diagonal action Γ y ( X × X, µ × µ ) has property (T) and is weakly mixing [PV08]. Theorem B (Cocycle superrigidity) . Let G be a simply connected l.c.s.c. group and Γ < G acountable dense subgroup. Assume that there exists a subgroup Γ < Γ such that g Γ g − ∩ Γ isdense in G , for all g ∈ Γ , and the translation action Γ y ( G, m G ) has property (T).Let Λ be a countable group and w : Γ × G → Λ a cocycle.Then there exist a homomorphism δ : Γ → Λ and a Borel map φ : G → Λ such that we have w ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G . As a consequence of Theorem B, we are able to describe all nonsingular actions that are SOE toΓ y ( G, m G ). More precisely, we prove that any such action is obtained from Γ y ( G, m G ) bytaking quotients and inducing, as explained in the following example: Example 1.5.
Let Γ y ( X, µ ) be a nonsingular action of a countable group Γ. For an equivalencerelation R on a set Y , we denote by R| Z := R ∩ ( Z × Z ) the restriction of R to a subset Z ⊂ Y . ADRIAN IOANA (1) Let Γ be a normal subgroup of Γ whose action on X has a fundamental domain, i.e. thereis a measurable set X such that X = ∪ g ∈ Γ gX and µ ( gX ∩ X ) = 0, for all g ∈ Γ \ { e } .Then the map π : X → Γ \ X given by π ( x ) = Γ x witnesses an isomorphism betweenthe restriction of R (Γ y X ) to X and R (Γ / Γ y Γ \ X ). As a consequence, the actionΓ / Γ y Γ \ X is stably orbit equivalent to Γ y X . Moreover, if Γ y ( X, µ ) is ergodic,infinite measure preserving and µ ( X ) = + ∞ , then these actions are orbit equivalent.(2) Let Γ be a countable group which contains Γ. Consider the action of Γ on Γ × X given by g · ( h, x ) = ( hg − , gx ) and the quotient space X := (Γ × X ) / Γ. Then the induced action Γ y X given by g · ( h, x )Γ = ( gh, x )Γ is stably orbit equivalent to Γ y X . Indeed, if X := (Γ × X ) / Γ, then the restriction of R (Γ y X ) to X is isomorphic to R (Γ y X ). Theorem C (OE superrigidity) . Let G be a simply connected l.c.s.c. group and Γ < G a countabledense subgroup. Assume that there exists a subgroup Γ < Γ such that g Γ g − ∩ Γ is dense in G ,for all g ∈ Γ , and the translation action Γ y ( G, m G ) has property (T).Let Λ y ( Y, ν ) be an arbitrary free ergodic nonsingular action of an arbitrary countable group Λ .Then Λ y Y is SOE to Γ y ( G, m G ) if and only if we can find a central discrete subgroup Γ < G which is contained in Γ , a subgroup Λ < Λ and a Λ -invariant measurable set Y ⊂ Y such that Γ / Γ y G/ Γ is conjugate to Λ y Y and the action Λ y Y is induced from Λ y Y . The above theorems show that several known rigidity phenomena for translation actions on compactgroups admit analogues in the case of translation actions on locally compact non-compact groups.This leads to the question: to what extent are these two classes of actions related. The followingresult provides an answer to this question.
Proposition D (Weak compactness) . Let G be a l.c.s.c. group and Γ < G a countable densesubgroup. Let A ⊂ G be a measurable set with < m G ( A ) < + ∞ . Endow A with the probabilitymeasure obtained by restricting and rescaling m G .Then the countable ergodic p.m.p. equivalence relation R (Γ y G ) | A is weakly compact. Here, we are using N. Ozawa and S. Popa’s notion of weak compactness for equivalence relations[OP07] (see Definition 7.1).1.4.
Orbit equivalence rigidity for general translation actions.
The results stated so farapply to “simple” translation actions Γ y ( G, m G ). Next, assuming that G is a real algebraic group,we present two rigidity results which apply to fairly general translation actions Γ y ( G/ Σ , m G/ Σ ). Theorem E (OE rigidity, II) . Let G be a connected real algebraic group with trivial center, Σ < G a discrete subgroup and Γ < G a countable dense subgroup. Assume that the translation action Γ y ( G, m G ) is strongly ergodic. Let H be a connected semisimple real algebraic group with trivialcenter, ∆ < H a discrete subgroup and Λ < H a countable subgroup.Then the actions Γ y ( G/ Σ , m G/ Σ ) and Λ y ( H/ ∆ , m H/ ∆ ) are SOE if and only if there exist atopological isomorphism δ : G → H and h ∈ H such that δ (Γ) = Λ and δ (Σ) = h ∆ h − . Remark 1.6.
In the context of Theorem E, assume additionally that G = SL n ( R ), for some n > < G is a lattice. Then Examples 1.7-1.9 below provide many examples of countabledense subgroups Γ < G such that the translation action Γ y G is strongly ergodic. On the otherhand, by [IS10, Theorem D], the action Γ y G/ Σ is rigid, in the sense of S. Popa [Po01]. Altogether,this shows that Theorem E applies to a large family of rigid actions.In the proof of Theorem E, we exploit the fact that the translation action Γ y G/ Σ is relatedto the simple translation action Γ y G , via the quotient map G → G/ Σ. Nevertheless, from the
RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 5 point of view of orbit equivalence, these actions are at opposite ends. Indeed, by Proposition D therestriction of R (Γ y G ) to any set of finite measure is weakly compact. On the other hand, sincethe action Γ y G/ Σ is rigid, the equivalence relation R (Γ y G/ Σ) is not weakly compact (thiscan be seen by using the ergodic-theoretic characterization of rigid actions from [Io09]).
Theorem F (OE rigidity, III) . Let G and H be connected real algebraic groups with trivial centers.Let K < G and
L < H be connected real algebraic subgroups such that ∩ g ∈ G gKg − = { e } and ∩ h ∈ H hLh − = { e } . Let Γ < G and Λ < H be countable dense subgroups, and assume that thetranslation actions Γ y ( G, m G ) and Λ y ( H, m H ) are strongly ergodic.Then the actions Γ y ( G/K, m
G/K ) and Λ y ( H/L, m
H/L ) are SOE if and only if there exists atopological isomorphism δ : G → H and h ∈ H such that δ (Γ) = Λ and δ ( K ) = hLh − . Theorems E and F both require that Γ < G is a countable dense subgroup. It would be interestingto decide whether similar results hold under the less restrictive assumption that the action Γ onthe respective homogeneous spaces is ergodic.1.5.
Strong ergodicity and property (T) for translation actions.
In view of the aboveresults, it is natural to wonder when are translation actions strongly ergodic or have property (T)?The next two results provide necessary conditions for a translation action Γ y ( G, m G ) to bestrongly ergodic or have property (T). Recall that a p.m.p. action G y σ ( X, µ ) of a l.c.s.c. group G has spectral gap if there does not exist a sequence of unit vectors ξ n ∈ L ( X, µ ) := L ( X, µ ) ⊖ C n →∞ sup g ∈ K k σ g ( ξ n ) − ξ n k = 0, for every compact set K ⊂ G . Proposition G.
Let G = G × G be a product of two l.c.s.c. groups and p : G → G be thequotient homomorphism. Let Γ < G be a lattice such that p (Γ) < G is dense. Consider thetranslation action Γ y ( G , m G ) given by g · x = p ( g ) x , for all g ∈ Γ , x ∈ G .Then we have the following:(1) If the action G y ( G/ Γ , m G/ Γ ) has spectral gap, then the translation action Γ y ( G , m G ) is strongly ergodic.(2) If G has property (T), then the translation action Γ y ( G , m G ) has property (T). Proposition H.
Let G = SL n ( R ) and K = SO n ( R ) , for some n > . Let Γ < G be a countablesubgroup which is not contained in K . Assume that Γ ∩ K is dense in K and the translation action Γ ∩ K y ( K, m K ) has spectral gap.Then the translation action Γ y ( G, m G ) is strongly ergodic. Next, we combine the previous propositions with known results on spectral gap and property (T)to provide concrete classes of translation actions that are strongly ergodic or have property (T).
Example 1.7.
Let Γ = SL n ( Z [ √ q ]) and G = SL n ( R ), where n, q > q is not asquare. Then the translation action Γ y ( G, m G ) is strongly ergodic if n >
2, and has property(T) if n >
3. Indeed, Γ is an irreducible lattice in G × G = SL n ( R ) × SL n ( R ) and the quotient( G × G ) / Γ is not compact. Then the action G y ( G × G ) / Γ has spectral gap, for any n > n >
3, then G has property (T) (see e.g. [Zi84]). Theassertion is now a consequence of Proposition G. Example 1.8.
Let Γ = SL n ( Z [ 1 p ]) and G be either SL n ( R ) or SL n ( Q p ), for a prime p . Then thetranslation action Γ y ( G, m G ) is strongly ergodic if n > n >
3. To explainhow this assertion follows from the literature, denote G = SL n ( R ) and G = SL n ( Q p ). Then Γ is ADRIAN IOANA an irreducible lattice in e G := G × G . Consider the unitary representation π : e G → U ( L ( e G/ Γ)).As is well-known, π is strongly L s , for some s : the function e G ∋ g → h π ( g ) ξ, η i is L s -integrable, forall ξ, η belonging to a dense subspace of L ( e G/ Γ) (see e.g. [GMO08, Theorem 1.11]). This impliesthat π ⊗ N is contained in a multiple of the left regular representation of e G , for all integers N > s G and G are non-amenable, the restrictions of π to G and G do not have almost invariantvectors, and therefore the actions of G and G on e G/ Γ have spectral gap. Moreover, if n > G and G have property (T). The assertion is now a corollary of Proposition G. Example 1.9.
Let G = SL n ( R ) and K = SO n ( R ), for n >
3. Let Γ < G be a countable subgroupwhich contains a matrix g ∈ G \ K as well as matrices g , ..., g l ∈ K that have algebraic entries andgenerate a dense subgroup of K . If n = 3, then the work of J. Bourgain and A. Gamburd [BG06]shows that the action Γ ∩ K y ( K, m K ) has spectral gap. Moreover, the very recent work [BdS14]implies that this statement holds for any n >
3. In combination with Proposition H, this showsthat the translation action Γ y ( G, m G ) is strongly ergodic.1.6. Applications.
By using the above examples, one obtains many concrete families of actionsto which Theorems A-F apply. Next, we present a sample of applications of our main results. Fora set of primes S , we denote by Z [ S − ] the subring of Q consisting of rational numbers whosedenominators have all prime factors from S . Corollary I.
Let m, n > and S, T be nonempty sets of primes. Then the translation actions SL m ( Z [ S − ]) y SL m ( R ) and SL n ( Z [ T − ]) y SL n ( R ) are SOE if and only if ( m, S ) = ( n, T ) .More generally, assume that Σ < SL m ( R ) , ∆ < SL n ( R ) are either (1) discrete subgroups, or(2) connected real algebraic subgroups. If the translation actions SL m ( Z [ S − ]) y SL m ( R ) / Σ and SL n ( Z [ T − ]) y SL n ( R ) / ∆ are SOE, then ( m, S ) = ( n, T ) . Remark 1.10.
Let us explain how in the “higher rank case”, the first part of Corollary I followsfrom R. Zimmer’s work [Zi84], provided that S and T are finite. We start with a general fact.Let Γ , Λ be lattices in two products of l.c.s.c. groups G = G × G , H = H × H , such that theleft translation actions Γ y G , Λ y H are SOE. Then [Ge03, Lemma 6] implies that the lefttranslation actions G y G/ Γ, H y H/ Λ are SOE.Let m, n > S, T finite sets of primes such that SL m ( Z [ S − ]) y SL m ( R ) and SL n ( Z [ T − ]) y SL n ( R ) are SOE. Assume that m > | S | >
2. Then G m,S := Q p ∈ S SL m ( Q p )has rank >
2. Recall that SL m ( Z [ S − ]) sits diagonally as a lattice in SL m ( R ) × G m,S . The previousparagraph then implies that the translation actions G m,S y ( SL m ( R ) × G m,S ) /SL m ( Z [ S − ]) and G n,T y ( SL n ( R ) × G n,T ) /SL n ( Z [ T − ]) are SOE. Applying [Zi84, Theorem 10.1.8] finally gives that G m,S ∼ = G n,T , and therefore ( m, S ) = ( n, T ). Remark 1.11.
Let m, n > q, r > SL m ( Z [ √ q ]) y SL m ( R ) and SL n ( Z [ √ r ]) y SL n ( R ) are SOE. Then the proof ofCorollary I shows that ( m, q ) = ( n, r ). If m >
3, then this conclusion also follows from R. Zimmer’sstrong rigidity theorem [Zi80]. Indeed, as in the previous remark, since SL m ( Z [ √ q ]) sits diagonallyas a lattice in SL m ( R ) × SL m ( R ), the actions SL m ( R ) y ( SL m ( R ) × SL m ( R )) /SL m ( Z [ √ q ]), SL n ( R ) y ( SL n ( R ) × SL n ( R )) /SL n ( Z [ √ r ]) are SOE. [Zi80, Theorem 4.3] now implies that m = n and the involved actions of SL m ( R ) are conjugate, from which it follows that q = r as well.The novelty here consists of being able to handle the case m = n = 2 and conclude that the actionsof SL ( R ) on ( SL ( R ) × SL ( R )) /SL ( Z [ √ q ]) are mutually non stably orbit equivalent, as q variesthrough all the positive integers that are not squares. RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 7
Let R be an ergodic countable measure preserving equivalence relation on an infinite standardmeasure space ( X, µ ). The automorphism group of R , denoted Aut( R ), consists of nonsingularisomorphisms θ : X → X such that ( θ × θ )( R ) = R , almost everywhere. Since R is measurepreserving and ergodic, there is mod( θ ) > θ ∗ µ = mod( θ ) µ . Then mod : Aut( R ) → R ∗ + is a homomorphism and its image F ( R ) := mod(Aut( R )) is called the fundamental group of R . Corollary J.
Let G = SL n ( R ) , for some n > . Let Γ < G be a countable dense subgroup whichcontains the center of G such that the translation action Γ y ( G, m G ) is strongly ergodic.Then F ( R (Γ y G )) = { } . In other words, any automorphism of R (Γ y G ) preserves m G . Remark 1.12.
If Γ = SL n ( Q ), then Example 1.8 implies the translation action Γ y ( G, m G ) isstrongly ergodic. Corollary J therefore implies that the fundamental group of R (Γ y G ) is trivial.This solves part (i) of [Ge03, Problem 15]. Now, if H = SL ( Q p ), then Example 1.8 also impliesthat the translation Γ y ( H, m H ) is strongly ergodic. The proof of [Io13, Corollary 9.2] shows that R (Γ y H ) has trivial fundamental group, which answers part (ii) of [Ge03, Problem 15].We end the introduction with an OE superrigidity result which describes all the actions that areSOE to P SL m ( Z [ S − ]) y P SL m ( R ) (or, equivalently, to SL m ( Z [ S − ]) y SL m ( R )), whenever m > S is a nonempty set of primes. More generally, we have: Corollary K.
Let m > be an integer, Σ < P SL m ( R ) a lattice, and S a nonempty set of primes.If Λ y ( Y, ν ) is an arbitrary free ergodic nonsingular action of an arbitrary countable group Λ , then(1) Λ y Y is SOE to the left translation action P SL m ( Z [ S − ]) y P SL m ( R ) if and only if wecan find a subgroup Λ < Λ and a finite normal subgroup N < Λ such that • Λ y Y is induced from some nonsingular action Λ y Y , and • Λ /N y Y /N is conjugate to P SL m ( Z [ S − ]) y P SL m ( R ) .(2) Λ y Y is SOE to the left translation action P SL m ( Z [ S − ]) y P SL m ( R ) / Σ if and only ifwe can find a subgroup Λ < Λ and a finite normal subgroup N < Λ such that • Λ y Y is induced from some nonsingular action Λ y Y , and • Λ /N y Y /N is conjugate to either P SL m ( Z [ S − ]) y P SL m ( R ) / Σ or the left-rightmultiplication action P SL m ( Z [ S − ]) × Σ y P SL m ( R ) given by ( g, σ ) · x = gxσ − . The second part of Corollary K shows that any free p.m.p. action Λ y Y which is SOE to the lefttranslation action P SL m ( Z [ S − ]) y P SL m ( R ) / Σ, must be virtually conjugate to it (in the senseof [Fu99, Definition 1.1]). This adds to the list of OE superrigid p.m.p. actions discovered recentlyin [Fu99, Po05, Po06, Ki06, Io08, PV08].1.7.
Acknowledgements.
I would like to thank Alireza Salehi-Golsefidy for helpful discussionson algebraic groups and in particular for pointing out Remark 10.2.2.
Preliminaries
Countable nonsingular equivalence relations.
Let (
X, µ ) be a standard measure space.An equivalence relation R on X is called countable nonsingular if it satisfies the following conditions: • every equivalence class [ x ] R = { y ∈ X | ( x, y ) ∈ R} is countable, • R is a Borel subset of X × X , and • the R -saturation ∪ x ∈ A [ x ] R of any null set A is also a null set. ADRIAN IOANA
If Γ y ( X, µ ) is a nonsingular action of a countable group Γ, then the orbit equivalence relation R (Γ y X ) := { ( x, y ) ∈ X × X | Γ x = Γ y } is a countable nonsingular equivalence relation. Conversely, J. Feldman and C.C. Moore provedthat every countable nonsingular equivalence relation arises this way [FM77].Let R be a countable nonsingular equivalence relation on ( X, µ ). We denote by [ R ] the full group of R consisting of all nonsingular isomorphisms θ : X → X such that ( θ ( x ) , x ) ∈ R , for almostevery x ∈ X . We also denote by Aut( R ) the automorphism group of R consisting of all nonsingularisomorphisms θ : X → X such that ( x, y ) ∈ R if and only if ( θ ( x ) , θ ( y )) ∈ R , for almost every( x, y ) ∈ R . Then [ R ] is a normal subgroup of Aut( R ). The quotient group is denoted by Out( R )and called the outer automorphism group of R .Finally, we say that R is measure preserving if every θ ∈ [ R ] preserves µ . If µ is a probabilitymeasure and R is measure preserving, then we say that R is probability measure preserving ( p.m.p. ).2.2. Strong ergodicity for countable equivalence relations.
Let R be a countable nonsin-gular equivalence relation on a standard probability space ( X, µ ). A sequence { A n } of measurablesubsets of X is said to be asymptotically invariant ( a.i. ) if lim n →∞ µ ( θ ( A n )∆ A n ) = 0, for all θ ∈ [ R ]. Definition 2.1. [CW81, Sc81] A countable nonsingular equivalence relation R is called stronglyergodic if any sequence { A n } of a.i. sets is trivial, i.e. satisfies lim n →∞ µ ( A n )(1 − µ ( A n )) = 0. Remark 2.2.
It can readily seen that strong ergodicity only depends on the measure class of µ .Thus, we say that an equivalence relation R on a standard measure space ( X, µ ) is strongly ergodicif it is strongly ergodic with respect to a probability measure µ which is equivalent to µ . It is easyto check that a nonsingular action Γ y ( X, µ ) is strongly ergodic (in the sense of Definition 1.3) ifand only if its orbit equivalence relation R (Γ y X ) is strongly ergodic.Next, we record the following result, whose proof is identical to the proof of [Io13, Lemma 2.7],although the latter is only stated in the case of p.m.p. actions. Lemma 2.3. [Io13] Let R be a countable nonsingular equivalence relation on a standard probabilityspace ( X, µ ) . Assume that R is strongly ergodic.Then for every ε > , we can find δ > and F ⊂ [ R ] finite such that if a Borel map ρ : X → Y with values into a standard Borel space Y satisfies µ ( { x ∈ X | ρ ( θ ( x )) = ρ ( x ) } ) > − δ , for all θ ∈ F , then there exists y ∈ Y such that µ ( { x ∈ X | ρ ( x ) = y } ) > − ε . Strong ergodicity for actions of locally compact groups.
The notion of strong ergodicityfor actions of countable groups has a natural extension to actions of locally compact groups.
Definition 2.4.
Let G y ( X, µ ) be a nonsingular ergodic action of a l.c.s.c. group G on a standardprobability space ( X, µ ). • A sequence { A n } of measurable subsets of X is said to be asymptotically invariant if itsatisfies lim n →∞ sup g ∈ K µ ( gA n ∆ A n ) = 0, for every compact set K ⊂ G . • The action G y ( X, µ ) is called strongly ergodic if any asymptotically invariant sequence { A n } is trivial, i.e. lim n →∞ µ ( A n )(1 − µ ( A n )) = 0. • A nonsingular action G y ( X, µ ) of a l.c.s.c. group G on a standard measure space ( X, µ )is strongly ergodic if it is strongly ergodic with respect to some (equivalently, to any) Borelprobability measure µ on X which is equivalent to µ .Next, we provide two constructions of strongly ergodic actions of locally compact groups. RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 9
Lemma 2.5. If G is a l.c.s.c. group, then the left translation action G y ( G, m G ) is stronglyergodic.Proof. Let µ be a Borel probability measure on G which is equivalent to m G . Let { A n } be asequence of measurable subsets of G such that sup g ∈ K µ ( gA n ∆ A n ) →
0, for every compact K ⊂ G .Our goal is to show that lim n →∞ µ ( A n )(1 − µ ( A n )) = 0.If β n = Z G µ ( g − A n ∆ A n ) d µ ( g ), then the dominated convergence theorem shows that lim n →∞ β n = 0.Let K ⊂ G be a compact set with m G ( K ) >
0. By Fubini’s theorem, for all n we have that Z K Z G | A n ( gx ) − A n ( x ) | d µ ( g ) d µ ( x ) Z G Z G | A n ( gx ) − A n ( x ) | d µ ( x ) d µ ( g ) = Z G µ ( g − A n ∆ A n ) d µ ( g ) = β n . Thus, for every n >
1, we can find x n ∈ K such that Z G | A n ( gx n ) − A n ( x n ) | d µ ( g ) β n µ ( K ) .If x n A n , then µ ( A n x − n ) β n µ ( K ) . On the other hand, if x n ∈ A n , then 1 − µ ( A n x − n ) β n µ ( K ) .In either case, we get that µ ( A n x − n )(1 − µ ( A n x − n ) β n µ ( K ) , for all n . Since β n →
0, we concludethat µ ( A n x − n )(1 − µ ( A n x − n )) → K is compact, after passing to a subsequence, we may assume that the sequence { x n } converges to some x ∈ K . Let m ′ G be a right invariant Haar measure on G . Then µ is absolutelycontinuous with respect to m ′ G , hence we can find f ∈ L ( G, m ′ G ) + such that d µ = f d m ′ G . Then | µ ( A n x − n ) − µ ( A n x − ) | Z G | f ( yx n ) − f ( yx ) | d m ′ G ( y ) and therefore | µ ( A n x − n ) − µ ( A n x − ) | → µ ( A n x − )(1 − µ ( A n x − )) →
0. This easilyimplies that µ ( A n )(1 − µ ( A n )) →
0, which finishes the proof. (cid:4)
Lemma 2.6.
Let G be a l.c.s.c. group. Let Γ < G be a lattice and Γ y α ( X, µ ) a nonsingular actionon a standard probability space ( X, µ ) . Assume that the induced action G y ˜ α ( G/ Γ × X, m G/ Γ × µ ) is strongly ergodic.Then α is strongly egodic.Proof. Let φ : G/ Γ → G be a Borel map such that φ ( x )Γ = x , for all x ∈ G/ Γ. Let w : G × G/ Γ → Γbe the cocycle given by w ( g, x ) = φ ( gx ) − gφ ( x ), for all g ∈ G and x ∈ G/ Γ. Then the induced action˜ α is given by g ( x, y ) = ( gx, w ( g, x ) y ), for all g ∈ G, x ∈ G/ Γ and y ∈ X . Denote ˜ µ = m G/ Γ × µ .Consider a sequence { A n } of measurable subsets of X such that µ ( gA n ∆ A n ) →
0, for all g ∈ Γ.Define ˜ A n = G/ Γ × A n . Then sup g ∈ K ˜ µ ( g ˜ A n ∆ ˜ A n ) →
0, for every compact set K ⊂ G . Since ˜ α isstrongly ergodic, we get that ˜ µ ( ˜ A n )(1 − ˜ µ ( ˜ A n )) →
0, which implies that µ ( A n )(1 − µ ( A n )) → (cid:4) Property (T) for actions and equivalence relations.
Next, we recall R. Zimmer’s notionof property (T) for actions and equivalence relations. Let (
X, µ ) be a standard measure space.Firstly, let Γ y ( X, µ ) be a nonsingular action of a countable group Γ and let G be a Polish group.A Borel map c : Γ × X → G is called a cocycle if it satisfies the relation c ( gh, x ) = c ( g, hx ) c ( h, x ),for all g, h ∈ Γ and almost every x ∈ X . Two cocycles c , c : Γ × X → G are called cohomologous if there exists a Borel map φ : X → G such that c ( g, x ) = φ ( gx ) c ( g, x ) φ ( x ) − , for all g ∈ Γ andalmost every x ∈ X . Definition 2.7. [Zi81] A nonsingular action Γ y ( X, µ ) is said to have property (T) if every cocycle c : Γ × X → U ( H ) into the unitary group of a Hilbert space H which admits a sequence of almostinvariant unit vectors necessarily has an invariant unit vector, where: • A sequence of almost invariant unit vectors is a sequence of Borel maps ξ n : X → H suchthat for almost every x ∈ X we have lim n →∞ k ξ n ( gx ) − c ( g, x ) ξ n ( x ) k = 0, for all g ∈ Γ, and k ξ n ( x ) k = 1, for all n > • An invariant unit vector is a Borel map η : X → H which satisfies η ( gx ) = c ( g, x ) η ( x ) and k η ( x ) k = 1, for all g ∈ Γ and almost every x ∈ X .Secondly, let R be a countable nonsingular equivalence relation on ( X, µ ) and G be a Polish group.A Borel map c : R → G is called a cocycle if for almost every x ∈ X we have c ( x, y ) c ( y, z ) = c ( x, z ),for all y, z ∈ [ x ] R . Two cocycles c , c : R → G are said to be cohomologous if there exists a Borelmap φ : X → G such that c ( x, y ) = φ ( x ) c ( x, y ) φ ( y ) − , for almost every x ∈ X and all y ∈ [ x ] R . Definition 2.8. [Zi81] A nonsingular equivalence relation R on ( X, µ ) is said to have property (T) if every cocycle c : R → U ( H ) into the unitary group of a Hilbert space H which admits a sequenceof almost invariant unit vectors necessarily has an invariant unit vector, where • A sequence of almost invariant unit vectors is a sequence of Borel maps ξ n : X → H suchthat for almost every x ∈ X we have that lim n →∞ k ξ n ( x ) − c ( x, y ) ξ n ( y ) k = 0, for all y ∈ [ x ] R ,and k ξ n ( x ) k = 1, for every n > • An invariant unit vector is a Borel map η : X → H which satisfies η ( x ) = c ( x, y ) η ( y ) and k η ( x ) k = 1, for almost every x ∈ X and all y ∈ [ x ] R . Remark 2.9.
Let R be an ergodic countable nonsingular equivalence relation on ( X, µ ). Then thefollowing hold:(1) If R has property (T), then it is of type II, i.e. there is an R -invariant measure ν on X which is equivalent to µ . This follows from the fact that any cocycle α : R → R ∗ + is trivial(see [Zi84, Theorem 9.1.1]).(2) If A ⊂ X is a non-null measurable subset, then R| A has property (T) if and only if R hasproperty (T).(3) If A ⊂ X is a non-null measurable subset, then R| A is strongly ergodic if and only if R isstrongly ergodic.(4) If R has property (T), then it is strongly ergodic. In the case when R is of type II , thiscan be deduced easily from [Pi04, Th´eor`eme 20]. In general, one can reduce to this case byusing the previous three facts.In the proof of Theorem B, we will need the following result asserting that if a p.m.p. equivalencerelation R has property (T), then any almost invariant vector is close to an invariant vector. Proposition 2.10. [Po86, Pi04] Let R be a countable ergodic p.m.p. equivalence relation on astandard probability space ( X, µ ) . Assume that R has property (T).Then there exist a constant κ > and a finite set F ⊂ [ R ] such that the following holds:Let c : R → U ( H ) be a cocycle, where H is a Hilbert space, and ξ : X → H a Borel map such that k ξ ( x ) k = 1 , for almost every x ∈ X . Then there is an invariant unit vector η : X → H such that Z X k η ( x ) − ξ ( x ) k d µ ( x ) κ X θ ∈ F Z X k ξ ( θ ( x )) − w ( θ ( x ) , x ) ξ ( x ) k d µ ( x ) . RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 11
Proposition 2.10 is an immediate consequence of [Pi04, Th´eor`eme 20]. Proposition 2.10 also followsfrom [Po86, Lemma 4.1.5], after noticing that a countable ergodic p.m.p. equivalence relation R has property (T) if and only if its von Neumann algebra L ( R ) has property (T) relative to L ∞ ( X )(in the sense of [Po86, Definition 4.1.3]).2.5. Algebraic groups and smooth actions.
The purpose of this subsection is to establishfollowing result about real algebraic groups that we will need in the proof of Theorem E.
Lemma 2.11.
Let G be a real algebraic group, H < G an R -subgroup, and denote N = ∩ g ∈ G gHg − .Then there are an integer n > , a G -invariant open conull subset Ω ⊂ ( G/H ) n and a Borel map π : Ω → G/N such that π ( gx ) = gπ ( x ) , for all g ∈ G and every x ∈ Ω .Moreover, for every g ∈ G \ N , the set { x ∈ G | gxH = xH } has measure zero. This result is most likely known, but for the lack of a reference, we include a proof. The proof ofLemma 2.11 relies on the fact that algebraic actions of algebraic groups on varieties are smooth.We begin by recalling the following (see [Zi84]):
Definition 2.12.
A Borel space X is called countably separated if there exists a sequence of Borelsets which separate points. A Borel action of a l.c.s.c. group G on a standard Borel space X iscalled smooth if the quotient Borel space X/G is countably separated.The next well-known theorem is due to Borel and Serre (see [Zi84, Theorem 3.1.3]).
Theorem 2.13.
If a real algebraic group G acts algebraically on an R -variety V , then the actionof G on V is smooth, where V is endowed with its natural Borel structure.Proof of Lemma 2.11. Let K = C . By a theorem of Chevalley (see [Zi84, Proposition 3.1.4]) we canfind a regular homomorphism π : G → GL r ( K ) and a point x ∈ K r such that H is the stabilizer of[ x ] ∈ P r − ( K ) in G . We may clearly assume that the set { π ( g ) x | g ∈ G } spans K r . Thus, we canfind h , ..., h r ∈ G such that the vectors { π ( h ) x, ..., π ( h r ) x } are linearly independent.Let Ω be the set of ( g , ..., g r ) ∈ G r such that the vectors { π ( g ) x, ..., π ( g r ) x } are linearly indepen-dent. Then Ω is a non-empty, Zariski open subset of G r . In particular, we get m G r ( G r \ Ω ) = 0,where m G r denotes the Haar measure on G r obtained by taking the r -fold product of m G with itself.Let n = r + 1 and Ω be the set of ( g , ..., g n ) ∈ G n such that the vectors { π ( g i ) x | i n, i = j } are linearly independent, for any j ∈ { , ..., n } . Then Ω is a G -invariant Zariski open subset of G n . Moreover, since m G r ( G r \ Ω ) = 0, we get that m G n ( G n \ Ω ) = 0.Next, we argue that if ( g , ..., g n ) ∈ Ω , then ∩ ni =1 g i Hg − i = N . Let h ∈ ∩ ni =1 g i Hg − i . Notice that π ( h ) stabilizes [ π ( g ) x ] , ..., [ π ( g n ) x ]. In other words, { π ( g ) x, ..., π ( g n ) x } are n = r + 1 eigenvectorsfor π ( h ) ∈ GL r ( K ). Since any r vectors from the set { π ( g ) x, ..., π ( g n ) x } are linearly independent,we get that π ( h ) = αI , for some α ∈ K \ { } . If g ∈ G , then π ( g − hg ) x = αx , hence g − hg ∈ H .This shows that h ∈ N , as claimed.Let Ω = { ( g H, ..., g n H ) | ( g , ..., g n ) ∈ Ω } . Since Ω is a G -invariant, open and conull subset of G n , we get that Ω is a G -invariant, open and conull subset of ( G/H ) n . Note that the action of G on ( G/H ) n descends to an action of G/N . Then the previous paragraph shows that the action
G/N y Ω is free, i.e. gx = x , for every g ∈ G/N , g = e and all x ∈ Ω.Now, if e H < e G are R -groups, then e G/ e H can be endowed with a natural R -variety structure on which e G (hence, any R -subgroup of e G ) acts R -algebraically (see [Zi84, Proposition 3.1.4]). In particular,( G/H ) n = G n /H n is an R -variety on which G acts R -algebraically. By applying Theorem 2.13 wederive that the action G y ( G/H ) n is smooth and hence it admits a Borel selector . More precisely, there exists a Borel map s : ( G/H ) n → ( G/H ) n satisfying s ( x ) ∈ Gx and s ( gx ) = s ( x ), for all g ∈ G and x ∈ ( G/H ) n (see [Ke95, Exercise 18.20 iii]).Since the action G/N y Ω is free, for every x ∈ Ω, there is a unique π ( x ) ∈ G/N such that x = π ( x ) s ( x ). The map π : Ω → G/N clearly satisfies π ( gx ) = gπ ( x ), for all g ∈ G/N and every x ∈ Ω. In order to finish the proof of the first assertion, it remains to prove that π is Borel. Tothis end, let F ⊂ G/N be a closed set.Let d be a metric on ( G/H ) n which gives the Hausdorff topology. Then f : ( G/H ) n → [0 , ∞ ) definedby f ( x ) = inf g ∈ F d ( x, gs ( x )) is Borel. Let x ∈ Ω such that f ( x ) = 0. Then there is a sequence { g m } m > in F such that g m s ( x ) → x , as m → ∞ . Since the action G y ( G/H ) n is smooth and thestabilizer of x is equal to N , the map G/N ∋ g → gx ∈ Gx is a homeomorphism [Zi84, Theorem2.1.14]. Thus, we can find g ∈ F such that g m → g , as m → ∞ . This implies that gs ( x ) = x andhence π ( x ) = g ∈ F . Altogether, it follows that { x ∈ Ω | π ( x ) ∈ F } = Ω ∩ { x ∈ ( G/H ) n | f ( x ) = 0 } is a Borel set. Since this holds for any closed set F ⊂ G/N , we get that π is Borel. This finishesthe proof of the first assertion.Finally, note that ∩ ni =1 g i Hg − i = N , for almost every ( g , ..., g n ) ∈ G n . It follows that if A ⊂ G haspositive measure, then ∩ g ∈ A gHg − = N , which implies the moreover assertion. (cid:4) Extensions of homomorphisms.
We end this section by recording a result about extendinghomomorphisms from a dense subgroup of a l.c.s.c. group to the whole group:
Lemma 2.14.
Let G be a l.c.s.c. group endowed with a Haar measure m G , and H be a Polishgroup. Let Γ < G be a dense subgroup and δ : Γ → H be a homomorphism. Assume that α : G → H is a Borel map such that for all g ∈ Γ we have that α ( gx ) = δ ( g ) α ( x ) , for almost every x ∈ G .Then δ extends to a continuous homomorphism δ : G → H and we can find h ∈ H such that α ( g ) = δ ( g ) h , for almost every g ∈ G . For a proof, see the proof of [Io13, Lemma 2.8].3.
Cocycle rigidity
The main goal of this section is to establish the following criterion for a cocycle for a translationaction Γ y G with values in a countable group Λ to be cohomologous to a homomorphism δ : Γ → Λ. Theorem 3.1.
Let G be a simply connected l.c.s.c. group and A a Borel subset with < m G ( A ) < + ∞ . Let Γ < G be a countable dense subgroup and denote R := R (Γ y G ) .Let Λ be a countable group and w : R → Λ be a cocycle. Suppose that there exist a constant C ∈ ( 3132 , and a neighborhood V of the identity in G such that (3.1) m G ( { x ∈ G | w ( α ( x ) t, xt ) = w ( α ( x ) , x ) } ) > C m G ( A ) , for all α ∈ [ R| A ] and every t ∈ V. Then we can find a homomorphism δ : Γ → Λ and a Borel map φ : G → Λ such that we have w ( gx, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G . The proof of this result is an adaptation of A. Furman’s proof of [Fu09, Theorem 5.21]. A mainingredient in the proof of Theorem 3.1 is the following immediate consequence of [Io08, Lemma 2.1]which provides a necessary condition for two cocycles to be cohomologous.
RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 13
Lemma 3.2.
Let R be a countable ergodic p.m.p. equivalence relation on a standard probabilityspace ( X, µ ) . Let Λ be a countable group and w , w : R → Λ be two cocycles.Let C ∈ ( 3132 , and assume that µ ( { x ∈ X | w ( α ( x ) , x ) = w ( α ( x ) , x ) } ) > C , for every α ∈ [ R ] . Then there exists a Borel map φ : X → Λ such that we have µ ( { x ∈ X | φ ( x ) = e } ) > and w ( x, y ) = φ ( x ) w ( x, y ) φ ( y ) − , for almost every ( x, y ) ∈ R .Proof . Let Γ < [ R ] be a countable subgroup which generates R [FM77]. Since R is ergodic,the action Γ y ( X, µ ) is ergodic. For i ∈ { , } , define a cocycle v i : Γ × X → Λ by letting v i ( g, x ) = w i ( gx, x ). Since C > and the action Γ y ( X, µ ) is ergodic, [Io08, Lemma 2.1] impliesthat we can find a Borel map φ : X → Λ such that v ( g, x ) = φ ( gx ) v ( g, x ) φ ( x ) − , for all g ∈ Γand almost every x ∈ X . Thus, w ( x, y ) = φ ( x ) w ( x, y ) φ ( y ) − , for almost every ( x, y ) ∈ R .Moreover, a close inspection of the proof of [Io08, Lemma 2.1] shows that φ verifies the following:there exists η ∈ L ( X × Λ , µ × c ) such that φ ( x ) is the unique λ ∈ Λ satisfying | η ( x, λ ) | > , and k η − X ×{ e } k √ − C < . Here, c denotes the counting measure on Λ. Since µ ( { x ∈ X | | η ( x, e ) | } ) Z X | η ( x, e ) − | d µ ( x ) k η − X ×{ e } k < , we conclude that µ ( { x ∈ X | φ ( x ) = e } ) > . (cid:4) Proof of Theorem 3.1.
As in the proof of [Fu09, Theorem 5.21], for any t ∈ G , we define anew cocycle w t : R →
Λ by letting w t ( x, y ) = w ( xt − , yt − ). Since C > , equation 3.1 implies thatthe restrictions of w , w t to R| A satisfy the assumptions of Lemma 3.2, for any t ∈ W := { t − | t ∈ V } .Thus, by applying Lemma 3.2, for any t ∈ W , we can find a Borel map φ t : A → Λ satisfying m G ( { x ∈ A | φ t ( x ) = e } ) > m G ( A ) and w ( xt − , yt − ) = φ t ( x ) w ( x, y ) φ t ( y ) − , for almost every( x, y ) ∈ R| A .Now, since Γ < G is dense, we can find a Borel map ψ : G → A such that ψ ( x ) = x , for all x ∈ A ,and ψ ( x ) ∈ Γ x , for almost every x ∈ G . For t ∈ W , we extend φ t : A → Λ to a map φ t : G → Λ byletting φ t ( x ) = w ( xt − , ψ ( x ) t − ) φ t ( ψ ( x )) w ( ψ ( x ) , x ). Then it is easy to check that(3.2) w ( xt − , yt − ) = φ t ( x ) w ( x, y ) φ t ( y ) − , for almost every ( x, y ) ∈ R and every t ∈ W. Next, we claim that whenever t, s, ts ∈ W we have that(3.3) φ ts ( x ) = φ t ( xs − ) φ s ( x ) , for almost every x ∈ G. To see this, let t, s ∈ W such that ts ∈ W , and define F ( x ) = φ ts ( x ) − φ t ( xs − ) φ s ( x ). Thenequation 3.2 implies that F ( x ) w ( x, y ) F ( y ) − = w ( x, y ), for almost every ( x, y ) ∈ R . This furthergives that the set B = { x ∈ G | F ( x ) = e } is R -invariant. Since m G ( { x ∈ A | φ t ( x ) = e } ) > m G ( A ),for every t ∈ W , we deduce that m G ( { x ∈ A | F ( x ) = e } ) > m G ( B ) >
0. Since R isergodic, we conclude that B = G , almost everywhere, which proves the claim.Since G is simply connected, the second part of the proof of [Fu09, Theorem 5.21] shows that wecan find a family of measurable maps { φ t : G → Λ } t ∈ G which extends the family { φ t : G → Λ } t ∈ W defined above in such a way that the identity 3.3 holds for every t, s ∈ G .By arguing exactly as in the end of the proof of [Fu09, Theorem 5.21] it follows that we can find ameasurable map φ : G → Λ such that(3.4) φ t ( x ) = φ ( xt − ) φ ( x ) − , for almost every ( x, t ) ∈ G × G. By combining equations 3.2 and 3.4 we get that φ ( xt − ) − w ( xt − , yt − ) φ ( yt − ) = φ ( x ) − w ( x, y ) φ ( y ),for almost every ( x, y ) ∈ R and almost every t ∈ G . Let g ∈ Γ and define L g : G → Λ by letting L g ( x ) = φ ( gx ) − w ( gx, x ) φ ( x ). Then we have that L g ( xt ) = L g ( x ), for almost every ( x, t ) ∈ G × G .This implies that we can find δ ( g ) ∈ Λ such that L g ( x ) = δ ( g ), for almost every x ∈ G . But then δ : Γ → Λ must be a homomorphism and the proof is finished. (cid:4)
We continue with the following consequence of Theorem 3.1 which will be a key ingredient in theproof of Theorem A.
Corollary 3.3.
Let G be a simply connected l.c.s.c. group and Γ < G be a countable dense subgroupsuch that the action Γ y G is strongly ergodic. Let Λ be a countable subgroup of a Polish group H and w : Γ × G → Λ be a cocycle. Assume that there exists a Borel map θ : G → H such that w ( g, x ) = θ ( gx ) θ ( x ) − , for all g ∈ Γ and almost every x ∈ G .Then there exist a homomorphism δ : Γ → Λ and a Borel map φ : G → Λ such that we have w ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G . Corollary 3.3 is a “locally compact analogue” of [Io13, Theorem 4.1] (see also [Io13, Remark 4.2]),with a very similar proof.
Proof.
Denote R := R (Γ y G ) and let v : R →
Λ be the cocycle given by v ( gx, x ) = w ( g, x ), forall g ∈ Γ and x ∈ G . Fix a Borel set A ⊂ G with 0 < m G ( A ) < + ∞ and let µ be the probabilitymeasure on A given by µ ( B ) = m G ( A ) − m G ( B ), for every B ⊂ A . Also, fix ε ∈ (0 , ).Since Γ y G is strongly ergodic, R and hence R| A is strongly ergodic. By Lemma 2.3, we canfind F ⊂ [ R| A ] finite and δ > ρ : A → Y into a standard Borel space Y satisfies µ ( { x ∈ A | ρ ( α ( x )) = ρ ( x ) } ) > − δ , for all α ∈ F , then there exists y ∈ Y such that µ ( { x ∈ A | ρ ( x ) = y } ) > − ε .Let α ∈ F . Then there is a Borel map γ : A → Γ such that α ( x ) = γ ( x ) x , for all x ∈ A . Hence,for all x ∈ A and t ∈ G we have that v ( α ( x ) t, xt ) = v ( γ ( x ) xt, xt ) = w ( γ ( x ) , xt ). Since γ and w take countably many values, it is easy to see that lim t → e µ ( { x ∈ A | w ( γ ( x ) , xt ) = w ( γ ( x ) , x ) } ) = 1.Equivalently, we have that lim t → e µ ( { x ∈ A | v ( α ( x ) t, xt ) = v ( α ( x ) , x ) } ) = 1. We can therefore find aneighborhood V of the identity in G such that(3.5) µ ( { x ∈ A | v ( α ( x ) t, xt ) = v ( α ( x ) , x ) } ) > − δ, for all α ∈ F and every t ∈ V. For every t ∈ V , we define a Borel map ρ t : G → H by letting ρ t ( x ) = θ ( x ) − θ ( xt ). Since ρ t ( α ( x )) = ρ t ( x ) ⇐⇒ v ( α ( x ) , x ) = v ( α ( xt ) , xt ), equation 3.5 rewrites as(3.6) µ ( { x ∈ A | ρ t ( α ( x )) = ρ t ( x ) } ) > − δ, for all α ∈ F and every t ∈ V. By combining 3.6 and the above consequence of strong ergodicity, for every t ∈ V , we can find y t ∈ H such that we have µ ( { x ∈ G | ρ t ( x ) = y t } ) > − ε . Hence, if α ∈ [ R| A ], then since α preserves µ , we get that µ ( { x ∈ G | ρ t ( α ( x )) = ρ t ( x ) } ) > − ε . From this we further derive that(3.7) µ ( { x ∈ A | v ( α ( x ) t, xt ) = v ( α ( x ) , x ) } ) > − ε for all α ∈ [ R| A ] and every t ∈ V. Since 1 − ε ∈ ( , δ : Γ → Λ and a Borel map φ : G → Λ such that v ( gx, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G . Thisclearly implies the conclusion. (cid:4) Approximately trivial cocycles.
We end this section by recalling [Io13, Lemma 4.2]. Notethat although [Io13, Lemma 4.2] is only stated for p.m.p. actions, its proof applies verbatim tononsingular actions.
RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 15
Lemma 3.4. [Io13] Let Γ y ( X, µ ) be a strongly ergodic nonsingular action of a countable group Γ on a standard probability space ( X, µ ) . Let H be a Polish group and w : Γ × X → H a cocycle.Assume that there exists a sequence of Borel maps { θ n : X → H } n > , such that for all g ∈ Γ wehave that lim n →∞ µ ( { x ∈ X | w ( g, x ) = θ n ( gx ) θ n ( x ) − } ) = 1 .Then there exists a Borel map θ : X → H such that w ( g, x ) = θ ( gx ) θ ( x ) − , for all g ∈ Γ andalmost every x ∈ X . Proof of Theorem A
A generalization of Theorem A.
We begin this section by proving the following moregeneral version of Theorem A:
Theorem 4.1.
Let G be a connected l.c.s.c. group and Γ < G a countable dense subgroup suchthat the action Γ y ( G, m G ) is strongly ergodic. Let H be a connected l.c.s.c. group and Λ < H a countable subgroup. Let A ⊂ G, B ⊂ H be non-negligible measurable sets and θ : A → B be anonsingular isomorphism such that θ (Γ x ∩ A ) = Λ θ ( x ) ∩ B , for almost every x ∈ A .Suppose that e G and e H are simply connected l.c.s.c. groups together with continuous onto homo-morphisms p : e G → G and q : e H → H such that ker( p ) < e G and ker( q ) < e H are discrete subgroups.Denote e Γ = p − (Γ) and e Λ = q − (Λ) .Then we can find a topological isomorphism δ : e G → e H such that δ ( e Γ) = e Λ , a Borel map φ : e G → Λ ,and h ∈ H such that θ ( p ( x )) = φ ( x ) q ( δ ( x )) h , for almost every x ∈ p − ( A ) .Moreover, if G and H have trivial centers, then we can find a topological isomorphism ¯ δ : G → H such that ¯ δ (Γ) = Λ , a Borel map φ : G → Λ , and h ∈ H such that θ ( x ) = φ ( x )¯ δ ( x ) h , for almostevery x ∈ A .Proof. Since the action Γ y G is ergodic, we can extend θ to a measurable map θ : G → H suchthat θ (Γ x ) ⊂ Λ θ ( x ), for almost every x ∈ G . Define ˜ θ : e G → H by letting ˜ θ ( x ) = θ ( p ( x )). Let w : e Γ × ˜ G → Λ be the cocycle given by the relation ˜ θ ( gx ) = w ( g, x )˜ θ ( x ).By Example 1.5 (1), the actions Γ y G and e Γ y e G are stably orbit equivalent. Since the actionΓ y G is strongly ergodic, we deduce that the action e Γ y e G is also strongly ergodic. Sinceapplying Corollary 3.3, we can find a homomorphism ρ : e Γ → Λ and a Borel map φ : e G → Λ suchthat w ( g, x ) = φ ( gx ) ρ ( g ) φ ( x ) − , for all g ∈ e Γ and almost every x ∈ e G .Define ˆ θ : e G → H by letting ˆ θ ( x ) = φ ( x ) − ˜ θ ( x ). Then ˆ θ ( gx ) = ρ ( g )ˆ θ ( x ), for all g ∈ e Γ and almostevery x ∈ e G . By Lemma 2.14, ρ extends to a continuous homomorphism ρ : e G → H and we canfind h ∈ H such that ˆ θ ( x ) = ρ ( x ) h , for almost every x ∈ e G . From this we get that(4.1) ˜ θ ( x ) = φ ( x ) ρ ( x ) h, for almost every x ∈ e G. We claim that ker( ρ ) is discrete in e G . Otherwise, we can find a sequence { g n } in ker( ρ ) \ { e } suchthat lim n →∞ g n = e . Since ker( p ) is discrete in e G , we may assume that p ( g n ) = e , for all n . Byusing 4.1 we derive that ˜ θ ( g n x ) = φ ( g n x ) φ ( x ) − ˜ θ ( x ), for almost every x ∈ e G . Since Λ is countable, m e G ( { x ∈ e G | φ ( g n x ) = φ ( x ) and p ( x ) , p ( g n x ) ∈ A } ) >
0, for n large enough. We would thus get that m e G ( { x ∈ e G | ˜ θ ( g n x ) = ˜ θ ( x ) and p ( x ) , p ( g n x ) ∈ A } ) >
0, for some n , contradicting the fact that therestriction of θ to A is 1-1 and p ( g n ) = e .Next, let us show that ρ is an onto open map. Let V be a neighborhood of e ∈ e G . Let W ⊂ e G be a compact subset such that W − W ⊂ V and m e G ( W ) >
0. Then ρ ( W ) ⊂ H is a compact subset. Moreover, since θ : A → B is a nonsingular isomorphism and p : e G → G is countable-to-1,we get that m H ( e θ ( W )) >
0. Since φ takes countably many values, by using 4.1 we deduce that m H ( ρ ( W )) >
0. By [Zi84, Lemma B.4] we derive that ρ ( W ) − ρ ( W ) and therefore ρ ( V ) contains aneighborhood of e ∈ H . This shows that ρ is an open map. In particular, ρ ( e G ) is an open subgroupof H . Since H is connected, we deduce that ρ ( e G ) = H .Altogether, we have that both q : e H → H and ρ : e G → H are covering homomorphisms. Since e H and e G are simply connected, by using the universality property of universal covering groups, wecan find a topological isomorphism δ : e G → e H such that q ◦ δ = ρ . Thus, equation 4.1 rewrites as(4.2) θ ( p ( x )) = ˜ θ ( x ) = φ ( x ) q ( δ ( x )) h, for almost every x ∈ e G. Finally, note that if g ∈ e Γ, then q ( δ ( g )) = ρ ( g ) ∈ Λ and hence δ ( g ) ∈ e Λ. Conversely, let g ∈ e G suchthat δ ( g ) ∈ ˜Λ. Then θ ( p ( gx )) = φ ( gx ) q ( δ ( gx )) ∈ Λ q ( δ ( x )) = Λ θ ( p ( x )), for almost every x ∈ e G .Note that by the construction of θ , for almost every x ∈ G we have that θ ( y ) ∈ Λ θ ( x ) ⇒ y ∈ Γ x .From this get we that p ( g ) p ( x ) = p ( gx ) ∈ Γ p ( x ), for almost every x ∈ e G . Therefore, p ( g ) ∈ Γ andhence g ∈ e Γ. This shows that δ ( e Γ) = e Λ and finishes the proof of the main assertion.For the moreover assertion, assume that G and H have trivial centers. Then ker( p ) = Z ( e G ) andker( q ) = Z ( e H ), and therefore δ descends to a topological isomorphism ¯ δ : G → H . It is now clearthat φ factors through the map p : e G → G , and the moreover assertion follows. (cid:4) Proof of Theorem A.
Since
G, H are simply connected, Theorem A follows by applyingTheorem 4.1 in the case e G = G , e H = H . (cid:4) The outer automorphism of R (Γ y G ) . Theorem 4.1 also allows us to compute the outerautomorphism group of R (Γ y G ). To state this precisely, for a l.c.s.c. group G and a subgroupΓ, we denote by Aut(Γ < G ) the group of topological automorphisms δ of G such that δ (Γ) = Γ. Corollary 4.2.
Let G be a connected l.c.s.c. group and Γ < G be a countable dense subgroupsuch that the action Γ y ( G, m G ) is strongly ergodic. Suppose that e G is a simply connected l.c.s.c.group together with an onto continuous homomorphism p : e G → G such that ker( p ) is discrete in e G . Denote R = R (Γ y G ) and e Γ = p − (Γ) .Consider the semidirect product L := e G ⋊ Aut ( e Γ < e G ) . For ˜ γ ∈ e Γ , denote by Ad (˜ γ ) ∈ Aut ( e Γ < e G ) the conjugation with ˜ γ . Then ∆ := { ( g, Ad (˜ γ )) | g ∈ e G, ˜ γ ∈ e Γ with p ( g ˜ γ ) = e } is a normal subgroupof L and we have the following:(1) Out ( R ) ∼ = L/ ∆ .(2) Assume additionally that every δ ∈ Aut ( e Γ < e G ) preserves m e G . Then F ( R ) = { } .Proof. (1) We begin the proof of this assertion with a claim. Let δ ∈ Aut( e Γ < e G ). Claim.
There exists θ δ ∈ Aut( R ) such that θ δ ( p ( x )) ∈ Γ p ( δ ( x )), for almost every x ∈ e G . Proof of the claim.
Since ker( p ) is discrete, we can find an open neighborhood V of e ∈ ˜ G such that m G ( V ) < ∞ and p is 1-1 on V − V ∪ δ ( V − V ). It follows that the map θ : p ( V ) → p ( δ ( V )) given by θ ( p ( x )) = p ( δ ( x )) is well-defined and 1-1. Since δ scales the Haar measure m G , δ and hence further θ is nonsingular. Moreover, for all x, y ∈ p ( V ) we have that Γ x = Γ y if and only if Γ θ ( x ) = Γ θ ( y ).Let us argue that θ extends to an automorphism θ δ ∈ Aut( R ). Assuming that this is the case, thensince θ δ ( p ( x )) = p ( δ ( x )), for all x ∈ V , it is easy to show that θ δ satisfies the claim. RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 17
To construct θ δ , we consider two cases. Firstly, assume that G is compact. Then m G is a finitemeasure, hence δ and θ preserve m G . Our claim now follows from the proof of [ ? , Lemma 2.2].Secondly, suppose that G is locally compact but not compact. Then m G is an infinite measure.Since m G ( V ) < ∞ , m G ( δ ( V )) < ∞ and R is ergodic, we can find sequences of disjoint measurablesubsets { X i } ∞ i =0 , { Y i } ∞ i =0 of G and of elements { α i } ∞ i =1 , { β i } ∞ i =1 in [ R ] such that • X = p ( V ) and Y = p ( δ ( V )). • X i = α i ( X ) and Y i = β i ( Y ), for all i > • G = ∪ i > X i = ∪ i > Y i , almost everywhere.We define θ δ by letting θ δ ( x ) = θ ( x ), if x ∈ X , and θ δ ( x ) = β i θα − i ( x ), if x ∈ X i , for i > (cid:3) Next, let ε : Aut( R ) → Out( R ) be the quotient homomorphism. If δ ∈ Aut( e Γ < e G ), then ε ( θ δ )only depends on δ (and not on the choices made in the proof of the claim). This allows to define ahomomorphism ρ : Aut( e Γ < e G ) → Out( R ) by letting ρ ( δ ) = ε ( θ δ ). Further, for g ∈ e G , we define θ g ∈ Aut( R ) by letting θ g ( x ) = xp ( g − ), for every x ∈ G . Then we consider the homomorphism ρ : e G → Out( R ) given by ρ ( g ) = ε ( θ g ). It is easy to check that ρ ( δ ) ρ ( g ) ρ ( δ ) − = ρ ( δ ( g )), forall δ ∈ Aut( e Γ < e G ) and every g ∈ e G .We can therefore define a homomorphism ρ : L → Out( R ) by letting ρ ( g, δ ) = ρ ( g ) ρ ( δ ) = ε ( θ g θ δ ),for every g ∈ e G and δ ∈ Aut( e Γ < e G ). Theorem 4.1 immediately gives that ρ is onto. If g ∈ e G and ˜ γ ∈ e Γ are such that p ( g ˜ γ ) = e , then θ g θ Ad(˜ γ ) ( p ( x )) ∈ Γ p (˜ γx ˜ γ − g − ) = Γ p ( x ), for almost every x ∈ e G . This shows that ∆ ⊂ ker( ρ ).Conversely, let ( g, δ ) ∈ ker( ρ ). Thus, θ g θ δ ∈ [ R ], hence p ( δ ( x ) g − ) ∈ Γ p ( x ), for almost every x ∈ e G .We derive that there exists γ ∈ Γ such that A = { x ∈ e G | p ( δ ( x ) g − ) = γp ( x ) } has positive measure.Since p ( δ ( xy − )) = γp ( xy − ) γ − , for all x, y ∈ A , the subgroup { x ∈ e G | p ( δ ( x )) = γp ( x ) γ − } of e G has positive measure. Since e G is connected, we conclude that p ( δ ( x )) = γp ( x ) γ − , for all x ∈ e G . Let ˜ γ ∈ e Γ such that p (˜ γ ) = γ . Then p ( δ ( x )) = p (˜ γx ˜ γ − ), for all x ∈ e G . Since ker( p ) isdiscrete and e G is connected, we deduce that δ = Ad(˜ γ ). Therefore, for almost every x ∈ A wehave that p (˜ γx ˜ γ − g − ) = p ( δ ( x ) g − ) = γp ( x ) = p (˜ γx ). We further get that p ( g ˜ γ ) = e and hence( g, δ ) = ( g, Ad(˜ γ )) ∈ ∆. This completes the proof of assertion (1).(2) Assume that every automorphism δ ∈ Aut( e Γ < e G ) preserves m e G . Then Ad(˜ γ ) preserves m e G ,for every ˜ γ ∈ e Γ. Since e Γ < e G is dense, we deduce that Ad( g ) preserves m e G , for every g ∈ e G . Thisimplies that e G is unimodular. It follows that the map e G ∋ x → δ ( x ) g ∈ e G preserves m e G , for every δ ∈ Aut( e Γ < e G ) and all g ∈ e G . Since the homomorphism ρ : L → Out( R ) defined above is onto,we get that every automorphism of R preserves m G and assertion (2) follows. (cid:4) Borel reducibility rigidity.
We end this section with an analogue of Theorem 4.1 for Borelreducibility. Let R , S be countable Borel equivalence relations on standard Borel spaces X, Y . Wesay that R is Borel reducible to S if there exists a Borel map θ : X → Y such that ( x, y ) ∈ R ifand only if ( θ ( x ) , θ ( y )) ∈ S . Theorem 4.3.
Let G be a connected l.c.s.c. group and Γ < G be a countable dense subgroup suchthat the action Γ y ( G, m G ) is strongly ergodic. Suppose that e G is a simply connected l.c.s.c.group together with an onto continuous homomorphism p : e G → G such that ker( p ) is discrete in e G . Denote e Γ = p − (Γ) . Let H be a l.c.s.c. group and Λ < H a countable subgroup.Then R (Γ y G ) is Borel reducible to R (Λ y H ) if and only if there exists a continuous homomor-phism δ : e G → H such that δ − (Λ) = e Γ . Proof.
To see the if part, assume that δ : e G → H is a continuous homomorphism such that δ − (Λ) = e Γ. Let r : G → e G be a Borel map such that p ( r ( x )) = x , for all x ∈ G . Then it is routineto check that θ : G → H given by θ ( x ) = δ ( r ( x )) is the desired Borel reduction.For the only if part, assume that there exists a Borel map θ : G → H such that x ∈ Γ y ifand only if θ ( x ) ∈ Λ θ ( y ). Then the proof of Theorem 4.1 shows that we can find a Borel map φ : e G → Λ, a continuous homomorphism δ : e G → H satisfying δ ( e Γ) ⊂ Λ, and h ∈ H such that θ ( p ( x )) = φ ( x ) δ ( x ) h , for almost every x ∈ e G . In order to finish the proof, we only need to arguethat δ − (Λ) ⊂ e Γ. To this end, let g ∈ G such that δ ( g ) ∈ Λ. Then for almost every x ∈ G , hencefor some x ∈ G , we have that θ ( p ( gx )) = φ ( gx ) δ ( gx ) h = ( φ ( gx ) δ ( g ) φ ( x ) − ) θ ( p ( x )) ∈ Λ θ ( p ( x )).This implies that p ( g ) p ( x ) = p ( gx ) ∈ Γ p ( x ) and therefore p ( g ) ∈ Γ, hence g ∈ ˜Γ, as desired. (cid:4) Proof of Theorem B
The proof of Theorem B follows closely the proof of [Fu09, Theorem 5.21]. The idea behind theproof is based on S. Popa’s deformation/rigidity theory. Roughly speaking, we exploit the tensionbetween the rigidity coming from the property (T) of the action Γ y G and the deformationassociated to the right multiplication action of G on itself.5.1. Proof of Theorem B.
Assume first that G is simply connected and the action Γ y ( G, m G )has property (T). Let w : Γ × G → Λ be a cocycle, where Λ is a countable group. Our goal is toshow that w is cohomologous to a homomorphism δ : Γ → Λ.Let A ⊂ G be a Borel subset with 0 < m G ( A ) < + ∞ . Since Γ y G has property (T), we get that R := R (Γ y G ) has property (T) and that R| A has property (T). Note that R| A preserves theprobability measure µ on A given by µ ( B ) = m G ( A ) − m G ( B ), for all measurable subsets B ⊂ A .By Proposition 2.10 we can find κ > F ⊂ [ R| A ] such that the following holds: if c : R → U ( H ) is a cocycle, where H is a Hilbert space, and ξ : A → H is a Borel map satisfying k ξ ( x ) k = 1, for almost every x ∈ A , then there exists an invariant unit vector η : A → H such that Z A k η ( x ) − ξ ( x ) k d µ ( x ) κ X θ ∈ F Z A k ξ ( θ ( x )) − c ( θ ( x ) , x ) ξ ( x ) k d µ ( x ) . Since η is an invariant vector and [ R| A ] preserves µ , it follows that for all θ ∈ [ R| A ] we have that(5.1) Z A k ξ ( θ ( x )) − c ( θ ( x ) , x ) ξ ( x ) k d µ ( x ) κ X θ ∈ F Z A k ξ ( θ ( x )) − c ( θ ( x ) , x ) ξ ( x ) k d µ ( x ) . Next, we let v : R →
Λ be given by v ( gx, x ) = w ( g, x ). For t ∈ G , we define a cocycle v t : R →
Λby letting v t ( x, y ) = v ( xt − , yt − ). Further, we define a cocycle c t : R → U ( ℓ (Λ)) by letting( c t ( x, y ) f )( λ ) = f ( v t ( x, y ) − λv ( x, y )) , for all ( x, y ) ∈ R , f ∈ ℓ (Λ) and λ ∈ Λ . Let ξ : A → ℓ (Λ) be given by ξ ( x ) = δ e , for all x ∈ A . Then we have c t ( x, y ) ξ ( y ) = δ v t ( x,y ) v ( x,y ) − ,for all x, y, t ∈ G , where { δ λ } λ ∈ Λ denotes the usual orthonormal basis of ℓ (Λ). Thus, by applyingequation 5.1 we get that for all θ ∈ [ R| A ] and every t ∈ G we have that(5.2) Z A k δ e − δ v t ( θ ( x ) ,x ) v ( θ ( x ) ,x ) − k d µ ( x ) κ X θ ∈ F Z A k δ e − δ v t ( θ ( x ) ,x ) v ( θ ( x ) ,x ) − k d µ ( x ) RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 19
Let ε ∈ (0 , √
232 ). Since Λ is countable, lim t → e µ ( { x ∈ A | v t ( θ ( x ) , x ) = v ( θ ( x ) , x ) } ) = 1, for any θ ∈ [ R| A ](see the proof of Corollary 3.3). Therefore, we can find a neighborhood V of e ∈ G such that(5.3) Z A k δ e − δ v t ( θ ( x ) ,x ) v ( θ ( x ) ,x ) − k d µ ( x ) ε κ | F | , for every t ∈ V and all θ ∈ F. By combining 5.2 and 5.3 we derive that for all t ∈ V and every θ ∈ [ R| A ], we have that √ µ ( { x ∈ A | v t ( θ ( x ) , x ) = v ( θ ( x ) , x ) } ) = Z A k δ e − δ v t ( θ ( x ) ,x ) v ( θ ( x ) ,x ) − k d µ ( x ) ε. Thus, we conclude that µ ( { x ∈ A | v t ( θ ( x ) , x ) = v ( θ ( x ) , x ) } ) > − ε √ t ∈ V and θ ∈ [ R| A ].Since 1 − ε √ > y ( G, m G ) has property (T). In general, theconclusion is obtained by combining this case with the following lemma. (cid:4) Lemma 5.1.
Let G be a connected l.c.s.c. group, Γ < G a countable subgroup, Γ < Γ a subgroup,and g ∈ Γ such that g Γ g − ∩ Γ is dense in G . Let Λ be a countable group, δ : Γ → Λ ahomomorphism, and w : Γ × G → Λ a cocycle such that w ( h, x ) = δ ( h ) , for all h ∈ Γ and almostevery x ∈ G .Then there is λ ∈ Λ such that w ( g, x ) = λ , for almost every x ∈ G .Proof. Let Γ = g Γ g − ∩ Γ and α : Γ → Γ given by α ( h ) = g − hg . If h ∈ Γ , then gα ( h ) = hg and the cocycle relation yields that w ( g, α ( h ) x ) δ ( α ( h )) = w ( gα ( h ) , x ) = w ( hg, x ) = δ ( h ) w ( g, x ),for almost every x ∈ G . Let S be the set of ( x, y ) ∈ G × G such that w ( g, x ) = w ( g, y ). Then thelast identity implies that S is invariant under the diagonal action of α (Γ ) on G × G . Since α (Γ )is dense in G , S must be invariant under the diagonal action of G on G × G .Therefore, since S is non-negligible, there is a non-negligible measurable set T ⊂ G such that S = { ( z, zt ) | z ∈ G, t ∈ T } , almost everywhere. As a consequence, the set T of all t ∈ G suchthat ( z, zt ) ∈ S , for almost every z ∈ G , is non-negligible. Since T < G is a subgroup and G isconnected, we derive that T = G , almost everywhere. Thus, S = G × G , almost everywhere, whichclearly implies the conclusion. (cid:4) Proof of Theorem C
Assume that a translation action Γ y G has property (T). In this section, we use Theorem Bto describe the actions that are SOE to Γ y G/ Σ, whenever Σ < G is a discrete subgroup. Inparticular, by applying this description in the case Σ = { e } , we deduce Theorem C. Theorem 6.1.
Let G be a simply connected l.c.s.c. group and Γ < G a countable dense subgroup.Assume that there exists a subgroup Γ < Γ such that g Γ g − ∩ Γ is dense in G , for all g ∈ Γ ,and the translation action Γ y ( G, m G ) has property (T). Let Σ < G be a discrete subgroup.Let Λ y ( Y, ν ) be a free ergodic nonsingular action of a countable group Λ which is SOE to Γ y ( G/ Σ , m G/ Σ ) .Then we can find a normal subgroup ∆ < Γ × Σ , a subgroup Λ < Λ , and a Λ -invariant Borelsubset Y ⊂ Y with ν ( Y ) > such that • ∆ is discrete in G × G , • the left-right multiplication action ∆ y G admits a measurable fundamental domain, • the action (Γ × Σ) / ∆ y G/ ∆ is conjugate to Λ y Y , and • the action Λ y Y is induced from Λ y Y .Proof. Let Λ y ( Y, ν ) be a free nonsingular action which is SOE to Γ y ( G/ Σ , m G/ Σ ). Since thelatter action is SOE to the action Γ × Σ y G , we deduce that Λ y ( Y, ν ) is SOE to Γ × Σ y G .Let A ⊂ G , B ⊂ Y be non-negligible measurable sets and θ : A → B a nonsingular isomorphismsuch that θ ((Γ × Σ) x ∩ A ) = Λ θ ( x ) ∩ B , for almost every x ∈ A . Since Γ y G is ergodic, we mayextend θ to a measurable map θ : G → Y such that θ ((Γ × Σ) x ) ⊂ Λ θ ( x ), for almost every x ∈ G .Define a cocycle w : Γ × G → Λ by the formula θ ( gx ) = w ( g, x ) θ ( x ), for all g ∈ Γ and almostevery x ∈ G . By applying Theorem B, we can find a homomorphism δ : Γ → Λ and a Borel map φ : G → Λ such that w ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G .The map ˜ θ : G → Y given by ˜ θ ( x ) = φ ( x ) − θ ( x ) therefore satisfies(6.1) ˜ θ ( gx ) = δ ( g )˜ θ ( x ) , for all g ∈ Γ and almost every x ∈ G . Claim . δ : Γ → Λ extends to a homomorphism δ : Γ × Σ → Λ such that ˜ θ ( gxσ − ) = δ ( g, σ )˜ θ ( x ),for all g ∈ Γ, σ ∈ Σ, and almost every x ∈ G . Proof of the claim.
Fix σ ∈ Σ. Since ˜ θ ( xσ − ) ∈ Λ˜ θ ( x ), for almost every x ∈ G , we can find a Borelmap v : G → Λ such that ˜ θ ( xσ − ) = v ( x )˜ θ ( x ), for almost every x ∈ G . By combining the fact thatthe actions of Γ and Σ on G commute with 6.1 and the freeness of the Λ-action, it follows that(6.2) v ( gx ) = δ ( g ) v ( x ) δ ( g ) − , for all g ∈ Γ and almost every x ∈ G .Define S to be the set of all ( x, y ) ∈ G × G such that v ( x ) = v ( y ). Equation 6.2 gives that S isinvariant under the diagonal action of Γ on G × G . Since Γ < G is dense, S must be invariant underthe diagonal action of G on G × G . By repeating the argument from the proof of Lemma 5.1, wederive that S = G × G , almost everywhere. This implies that v : G → Λ is a constant function. Ifwe denote this constant by δ ( σ ), then ˜ θ ( xσ − ) = δ ( σ )˜ θ ( x ), for almost every x ∈ G . It is then clearthat δ : Σ → Λ is a homomorphism. Moreover, by equation 6.2 we get that δ ( σ ) commutes with δ (Γ). Altogether, the claim follows. (cid:3) Let ∆ := ker( δ ). Assume by contradiction that ∆ is not discrete in G × G . Since Σ < G is discrete,it follows there is a sequence g n ∈ Γ \{ e } such that lim n →∞ g n = e and ( g n , e ) ∈ ∆, for all n . Fix a Borelset A ⊂ A with 0 < m G ( A ) < ∞ . Then ˜ θ ( g n x ) = ˜ θ ( x ), for all n , and almost every x ∈ A . Since φ takes countably many values, we also have that lim n →∞ m G ( { x ∈ A | φ ( g n x ) = φ ( x ) } ) = m G ( A ).By combining these facts, we get that m G ( { x ∈ A | θ ( g n x ) = θ ( x ) } ) >
0, for n large enough. Thiscontradicts the fact that the restriction of θ to A is 1 − ρ := θ − : B → A . Since ρ (Λ y ∩ B ) = (Γ × Σ) ρ ( y ) ∩ A , for almost every y ∈ B ,and the action Λ y Y is ergodic, we may extend ρ to a measurable map ρ : Y → G such that ρ (Λ y ) ⊂ (Γ × Σ) ρ ( y ), for almost every y ∈ Y . Then ρ (Λ˜ θ ( x )) = ρ (Λ θ ( x )) ⊂ (Γ × Σ) x , for almostevery x ∈ G , and the rest of the assertions are a consequence of the following lemma. (cid:4) Lemma 6.2.
Let Γ y ( X, µ ) be a nonsingular action of a countable group Γ and Λ y ( Y, ν ) be a free nonsingular action of a countable group Λ . Assume that there exist nonsingular maps θ : X → Y , ρ : Y → X and a group homomorphism δ : Γ → Λ such that ρ (Λ θ ( x )) ⊂ Γ x , for almostevery x ∈ X , and θ ( gx ) = δ ( g ) θ ( x ) , for all g ∈ Γ and almost every x ∈ X .Define Γ := ker( δ ) , Λ := δ (Γ) and Y := θ ( X ) .Then the action Γ y X admits a measurable fundamental domain, the action Γ / Γ y X/ Γ isconjugate to Λ y Y , and the action Λ y Y is induced from Λ y Y . RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 21
Proof.
Consider the nonsingular map τ := ρ ◦ θ : X → X . Let X ⊂ X be a maximal measurableset such that µ ( gX ∩ X ) = 0, for all g ∈ Γ \ { e } . Define X = ∪ g ∈ Γ gX . If X = X , then X isa fundamental domain for Γ y X . Assume by contradiction that µ ( X \ X ) >
0. Since τ ( x ) ∈ Γ x ,for almost every x ∈ X , there is a non-negligible subset X ⊂ X \ X such that τ | X is 1-1. Notethat τ ( gx ) = τ ( x ), for all g ∈ Γ and almost every x ∈ X . We deduce that µ ( gX ∩ X ) = 0, forall g ∈ Γ \ { e } . Since X ⊂ X \ X and X is Γ -invariant, we get that X = X ∪ X also satisfiesthat µ ( gX ∩ X ) = 0, for all g ∈ Γ \ { e } . This contradicts the maximality of X .Since the action Γ y X has a measurable fundamental domain, the quotient space X/ Γ endowedwith the push-forward of µ is a standard measure space. Since θ ( gx ) = θ ( x ), for all g ∈ Γ andalmost every x ∈ X , letting ¯ θ : X/ Γ → Y given by ¯ θ (Γ x ) = θ ( x ) defines an onto nonsingularmap. Moreover, ¯ θ is 1-1. Indeed, if x, y ∈ X are such that θ ( x ) = θ ( y ), then τ ( x ) = τ ( y ), henceΓ x = Γ y . Let g ∈ Γ such that y = gx . Since θ ( x ) = θ ( y ) = δ ( g ) θ ( x ) and the action Λ y Y is free,we deduce that δ ( g ) = e . Hence g ∈ Γ and therefore Γ x = Γ y . This shows that θ is 1-1. It isnow clear that ¯ θ is a conjugacy between Γ / Γ y X/ Γ and Λ y Y .To see that the action Λ y Y is induced from Λ y Y , let h ∈ Λ such that ν ( hY ∩ Y ) > x , x ∈ X such that y = θ ( x ) and h − y = θ ( x ). But then we get that θ ( x ) ∈ Λ θ ( x ) and by applying ρ we deduce that x ∈ Γ x . Let g ∈ Γ such that x = gx . Thus, h − y = θ ( x ) = δ ( g ) θ ( x ) = δ ( g ) y . This shows that h = δ ( g ) − ∈ Λ , which finishes the proof. (cid:4) Proof of Theorem C.
If ∆ < Γ is a normal subgroup which is discrete in G , then sinceΓ < G is dense and G is connected, ∆ must be central in G . Using this fact, Theorem C followsimmediately by applying Theorem 6.1 in the case Σ = { e } . (cid:4) Proof of Proposition D
In preparation for the proof of Proposition D, we recall the notion of weak compactness for countablep.m.p. equivalence relations. In [OP07, Definition 3.1], N. Ozawa and S. Popa defined the notion ofweak compactness for p.m.p. actions Γ y ( X, µ ). In [OP07, Proposition 3.4] they established thatif the action Γ y ( X, µ ) is weakly compact, then the action of the full group of R (Γ y X ) on X is also weakly compact. Thus, one can define weak compactness for countable p.m.p. equivalencerelations R by insisting that the associated action of the full group [ R ] is weakly compact: Definition 7.1. [OP07] A countable p.m.p. equivalence relation R on a standard probability space( X, µ ) is said to be weakly compact if there exists a net of vectors η n ∈ L ( X × X, µ ⊗ ) such that η n > k η n k = 1, for all n , and the following conditions are satisfied:(1) lim n k η n − ( u ⊗ ¯ u ) η n k = 0, for all u ∈ U ( L ∞ ( X )).(2) lim n k η n − η n ◦ ( θ × θ ) k = 0, for all θ ∈ [ R ].(3) lim n h ( v ⊗ η n , η n i = lim n h (1 ⊗ v ) η n , η n i = Z X v d µ , for all v ∈ L ∞ ( X ).Here, for u, v ∈ L ∞ ( X ), the function u ⊗ ¯ v ∈ L ∞ ( X × X ) is given by ( u ⊗ ¯ v )( x, y ) = u ( x ) v ( y ). Also, µ ⊗ denotes the product measure µ ⊗ µ on X × X . Note that the above conditions are preciselyconditions (1),(2) and (3’) from [OP07, Definition 3.1] for the action [ R ] y ( X, µ ).7.1.
Proof of Proposition D.
Denote R = R (Γ y G ) and let A ⊂ G be a Borel set with0 < m G ( A ) < + ∞ . Then µ = m G ( A ) − m G is a Haar measure of G such that µ ( A ) = 1.Our goal is to show that R| A is weakly compact. Note first that we may assume that A is an opensubset of G such that ¯ A is compact. This is because we can find an open set B ⊂ G such that ¯ B is compact and m G ( A ) = m G ( B ). Then since R is ergodic and preserves m G , there exists θ ∈ [ R ]such that θ ( A ) = B , hence R| A ∼ = R| B .Since G is second countable, we can find a left invariant compatible metric d on G . For x ∈ G and r >
0, we denote by B r ( x ) = { y ∈ G | d ( x, y ) < r } the open ball of radius r centered at x . We alsolet B r = B r ( e ). Notice that µ ( B r ( x )) = µ ( xB r ) = µ ( B r ).For ε >
0, we define S ε = { ( x, y ) ∈ A × A | d ( x, y ) < ε } . Then µ ⊗ ( S ε ) >
0. Indeed, otherwise wewould get that µ ( B ε ( x ) ∩ A ) = 0, for almost every x ∈ A . Since A is open, this would imply thatthere exists x ∈ A and ε ′ > µ ( B ε ′ ( x )) = 0, which contradicts the fact that µ is a Haarmeasure of G . Thus, we may further define η ε := 1 S ε p µ ⊗ ( S ε ) ∈ L ( A × A, µ ⊗ )Then η ε > k η ε k = 1, for all ε >
0. We will show that the net ( η ε ) verifies conditions(1)-(3) from Definition 7.1. Firstly, we verify condition (3). To this end, for ε >
0, we define ξ ε : A → [0 , ∞ ) by letting ξ ε ( x ) = Z A η ε ( x, y ) d µ ( y ). Claim 1 . We have that lim ε → k ξ ε k ∞ = 1 and lim ε → ξ ε ( x ) = 1, for almost every x ∈ A . Proof Claim 1.
We define a function r : A → (0 , ∞ ) by letting r ( x ) = sup { r > | B r ( x ) ⊂ A } . Since A is open, r is a well-defined continuos function and B r ( x ) ( x ) ⊂ A , for all x ∈ A .Let n >
1. Then there exists ε n > A n := { x ∈ A | r ( x ) > ε n } satisfies µ ( A n ) > − − n − .Let ε ∈ (0 , ε n ]. Then for all x ∈ A n we have that B ε ( x ) ⊂ A and hence µ ⊗ ( S ε ) = Z A µ ( B ε ( x ) ∩ A ) d µ ( x ) > Z A n µ ( B ε ( x )) d µ ( x ) = µ ( B ε ) µ ( A n ) > (1 − − n − ) µ ( B ε ) . As a consequence, for every x ∈ A we have that ξ ε ( x ) = µ ⊗ ( S ε ) − µ ( B ε ( x ) ∩ A ) µ ⊗ ( S ε ) − µ ( B ε ) < (1 − − n − ) − < − n . This shows that(7.1) k ξ ε k ∞ < − n , for all ε ∈ (0 , ε n ] . On the other hand, we have that µ ⊗ ( S ε ) = Z A µ ( B ε ∩ A ) d µ ( x ) µ ( B ε ). Thus, if x ∈ A n , then ξ ε ( x ) = µ ⊗ ( S ε ) − µ ( B ε ( x ) ∩ A ) = µ ⊗ ( S ε ) − µ ( B ε ) >
1. Hence, we get that(7.2) ξ ε ( x ) > , for all ε ∈ (0 , ε n ] and every x ∈ A n . Since µ ( A n ) > − − n − , for all n , it is easy to see that 7.1 and 7.2 together imply the claim. (cid:3) If v ∈ L ∞ ( A ), then h ( v ⊗ η ε , η ε i = Z A v ( x ) ξ ε ( x ) d µ ( x ). By combining Claim 1 and the Lebesguedominated convergence theorem, we get that lim ε → h ( v ⊗ η ε , η ε i = Z A v ( x )d µ ( x ). Since η ε is sym-metric, we have that h (1 ⊗ v ) η ε , η ε i = h ( v ⊗ η ε , η ε i and altogether condition (3) follows.Towards showing that the net ( η ε ) verifies conditions (1) and (2), we first establish the following: Claim 2.
We have that lim ε → k ( v ⊗ η ε − (1 ⊗ v ) η ε k = 0, for all v ∈ L ∞ ( A ). Proof of Claim 2.
Let C be the set of functions v ∈ L ∞ ( A ) for which there exists a continuousfunction ˜ v : ¯ A → C such that v = ˜ v | A . Then C is k . k -dense in L ∞ ( A ). On the other hand, RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 23 condition (3) implies that lim ε → k ( v ⊗ η ε k = lim ε → k (1 ⊗ v ) η ε k = k v k , for every v ∈ L ∞ ( A ). Thus,in order to prove the claim, it suffices to show that it holds for every v ∈ C .Let v ∈ C and ˜ v be a continuous extension of v to ¯ A . Let δ >
0. Since ˜ v is continuous and ¯ A iscompact, ˜ v is uniformly continuous. It follows that we can find ε > | v ( x ) − v ( y ) | < δ ,for all x, y ∈ A such that d ( x, y ) ε . From this we deduce that for all ε ∈ (0 , ε ] we have k ( v ⊗ η ε − (1 ⊗ v ) η ε k = µ ⊗ ( S ε ) − Z S ε | v ( x ) − v ( y ) | d µ ⊗ ( x, y ) < δ . Since δ > ε → k ( v ⊗ η ε − (1 ⊗ v ) η ε k = 0. (cid:3) Now, if u ∈ U ( L ∞ ( A )), then by Claim 2 we get k η ε − ( u ⊗ ¯ u ) η ε k = k (1 ⊗ u ) η ε − ( u ⊗ η ε k → ε →
0, which proves condition (1).Finally, in order to prove condition (2), let θ ∈ [ R| A ]. Then we can find a Borel map φ : A → Γsuch that θ ( x ) = φ ( x ) x , for almost every x ∈ A . Notice that if x, y ∈ A and φ ( x ) = φ ( y ), then wehave d ( θ ( x ) , θ ( y )) = d ( x, y ). Using this observation we get that if ε > k η ε − η ε ◦ ( θ × θ ) k = µ ⊗ ( S ε ) − Z X × X | S ε ( x, y ) − S ε ( θ ( x ) , θ ( y )) | d µ ⊗ ( x, y ) µ ⊗ ( S ε ) − µ ⊗ ( { ( x, y ) ∈ S ε | φ ( x ) = φ ( y ) } ) . For g ∈ Γ, denote A g = { x ∈ A | φ ( x ) = g } . Then we have(7.4) µ ⊗ ( { ( x, y ) ∈ S ε | φ ( x ) = φ ( y ) } ) X g ∈ Γ µ ⊗ ( { ( x, y ) ∈ S ε | x ∈ A g , y / ∈ A g } ) µ ⊗ ( S ε ) X g ∈ Γ k (1 A g ⊗ η ε − (1 ⊗ A g ) η ε k The combination of 7.3 and 7.4 further implies that(7.5) k η ε − η ε ◦ ( θ × θ ) k X g ∈ Γ k (1 A g ⊗ η ε − (1 ⊗ A g ) η ε k , for all ε > . Since lim ε → k ξ ε k ∞ = 1 by Claim 1, we can find ε such that k ξ ε k ∞
2, for all ε ∈ (0 , ε ]. Let v ∈ L ∞ ( A ) and ε ∈ (0 , ε ]. Then k ( v ⊗ η ε k = Z A × A | v | ξ ε d µ k v k . Since η ε is symmetric, weget that k (1 ⊗ v ) η ε k = k ( v ⊗ η ε k k v k . As a consequence, we have that(7.6) k ( v ⊗ η ε − (1 ⊗ v ) η ε k k v k , for all v ∈ L ∞ ( A ) and every ε ∈ (0 , ε ] . Let δ >
0. Then we can find a finite set F ⊂ Γ such that P g ∈ Γ \ F µ ( A g ) δ
16 . In combinationwith equation 7.6 we further get(7.7) X g ∈ Γ \ F k (1 A g ⊗ η ε − (1 ⊗ A g ) η ε k X g ∈ Γ \ F k A g k δ , for all ε ∈ (0 , ε ] . Next, by using condition (1), we can find ε ∈ (0 , ε ] such that(7.8) X g ∈ F k (1 A g ⊗ η ε − (1 ⊗ A g ) η ε k δ , for all ε ∈ (0 , ε ] . Finally, the combination of equations 7.5, 7.7 and 7.8 gives that k η ε − η ε ◦ ( θ × θ ) k δ , for all ε ∈ (0 , ε ]. Since δ > η ε ) satisfies condition (2). (cid:4) Proof of Theorem E
This section is devoted to the proof of Theorem E. In fact, we prove the following more generaland more precise version of Theorem E:
Theorem 8.1.
Let G be a connected l.c.s.c. with trivial center. Assume that there is a l.c.s.c.simply connected group e G together with a continuous onto homomorphism p : e G → G such that ker( p ) < e G is discrete. Let Σ < G be a discrete subgroup and Γ < G a countable dense subgroup.Assume that the translation action Γ y ( G, m G ) is strongly ergodic. Let ¯ H be a semisimple realalgebraic group and denote H = ¯ H/Z ( ¯ H ) . Let ∆ < H be a discrete subgroup and Λ < H a countablesubgroup.Let A ⊂ G/ Σ and B ⊂ H/ ∆ be non-negligible measurable sets, and θ : A → B be a nonsingularisomorphism such that θ (Γ x ∩ A ) = Λ θ ( x ) ∩ B , for almost every x ∈ A .Then we can find a Borel map φ : G/ Σ → Λ , a topological isomorphism δ : G → H and h ∈ H suchthat δ (Γ) = Λ , δ (Σ) = h ∆ h − and θ ( x ) = φ ( x ) δ ( x ) h ∆ , for almost every x ∈ A . Before proving Theorem 8.1, we establish the following elementary result:
Lemma 8.2.
Let G be a connected l.c.s.c. group, Γ < G a countable subgroup and Λ < G a discretesubgroup. If Γ ∩ Λ contains no non-trivial central element of G , then the action Γ y G/ Λ is free.Proof. Let g ∈ Γ such that the set { x ∈ G | gx Λ = x Λ } has positive measure. Since Λ is countable,we can find h ∈ Λ such that the set A = { x ∈ G | gx = xh } has positive measure. Since A haspositive measure, AA − contains a neighborhood of e ∈ G . Since g commutes with AA − and G is connected, we get that g belongs to the center Z ( G ) of G . Since gx = xh , for some x ∈ G , itfollows that g = h ∈ Γ ∩ Λ ∩ Z ( G ). Therefore, g = e , as claimed. (cid:4) Proof of Theorem 8.1.
Since the action Γ y G/ Σ is ergodic, we may extend θ to acountable-to-1 map θ : G/ Σ → H/ ∆ such that θ (Γ x ) ⊂ Λ θ ( x ), for almost every x ∈ G/ Σ.Denote by π : G → G/ Σ the quotient. Then θ ( π ( gx )) ∈ Λ θ ( π ( x )), for all g ∈ Γ and almost every x ∈ G . Since H has trivial center, by Lemma 8.2, the action Λ y H/ ∆ is free. Thus, we can definea cocycle W : Γ × G → Λ through the formula θ ( π ( gx )) = W ( g, x ) θ ( π ( x )). The freeness of theΛ-action also implies that W factors through the map Γ × G → Γ × G/ Σ.Let r : H/ ∆ → H be a Borel map such that r ( x )∆ = x , for all x ∈ H/ ∆. Further, we defineΘ : G → H by letting Θ( x ) = r ( θ ( π ( x ))). Then for all g ∈ Γ and almost every x ∈ G/ Σ we haveΘ( gx )∆ = r ( θ ( π ( gx )))∆ = θ ( π ( gx )) = W ( g, x ) θ ( π ( x )) = W ( g, x )Θ( x )∆ . We can therefore find a Borel map v : Γ × G → ∆ such that(8.1) Θ( gx ) = W ( g, x )Θ( x ) v ( g, x ) , for all g ∈ Γ and almost every x ∈ G/ Σ . The core of the proof is divided in two parts.
The first part of the proof . In this part we analyze the cocycle W and show that there existsa Borel map α : G → H such that W ( g, x ) = α ( gx ) α ( x ) − , for all g ∈ Γ and almost every x ∈ G .To this end, for a subset S ⊂ H , we denote by C ( S ) its centralizer in H . We denote by A theset of pairs ( h, k ) ∈ H × H such that C ( { h, k } ) = { e } . Since H is a connected, semisimple real RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 25
Lie group, [Wi02, Theorem 4] implies that the subgroup generated by h and k is dense in H , foralmost every ( h, k ) ∈ H × H . Since H has trivial center, for any such pair ( h, k ) we have that C ( { h, k } ) = { e } . Altogether, we deduce that A is conull in H × H .Next, for t ∈ G , we define a Borel map ρ t : G → H by letting ρ t ( x ) = Θ( xt )Θ( x ) − . Claim 1.
For almost every ( s, t ) ∈ G × G we have that ( ρ s ( x ) , ρ t ( x )) ∈ A , for almost every x ∈ G . Proof of Claim 1.
Let x ∈ G and define A x = { ( h Θ( x ) , k Θ( x )) | ( h, k ) ∈ A} . Since A is conullin H × H , A x is also conull in H × H . Since θ : G/ Σ → H/ ∆ is nonsingular, Θ : G → H isalso nonsingular: if B ⊂ H satisfies m H ( B ) = 0, then m G (Θ − ( B )) = 0. By combining these twofacts we conclude that the set { ( a, b ) ∈ G × G | (Θ( a ) , Θ( b )) ∈ A x } is conull. Further, we get that { ( s, t ) ∈ G × G | ( ρ s ( x ) , ρ t ( x )) ∈ A} = { ( s, t ) ∈ G × G | (Θ( xs ) , Θ( xt )) ∈ A x } is conull in G × G . Sincethis holds for all x ∈ G , we derive that the set { ( s, t, x ) ∈ G × G × G | ( ρ s ( x ) , ρ t ( x )) ∈ A} is conullin G × G × G . The claim now follows by applying Fubini’s theorem. (cid:3) Let µ be a Borel probability measure on G which is equivalent to m G . Claim 2.
There exists a sequence of Borel maps α n : G → H such that for all g ∈ Γ we havelim n →∞ µ ( { x ∈ G | α n ( gx ) = W ( g, x ) α n ( x ) } ) = 1. Proof of Claim 2.
Let ε > F ⊂ Γ finite be arbitrary. In order to prove the claim, it sufficesto find a Borel map α : G → H such that µ ( { x ∈ G | α ( gx ) = W ( g, x ) α ( x ) } ) > − ε , for all g ∈ F .Since µ is Γ-quasi-invariant, we can find ε ∈ (0 , ε B ⊂ G satisfies µ ( B ) > − ε , then µ ( g − B ) > − ε g ∈ F . Since the action Γ y G is strongly ergodic,Lemma 2.3 provides δ > S ⊂ Γ such that the following holds: if ρ : G → Y is aBorel map into a standard Borel space Y which satisfies µ ( { x ∈ G | ρ ( gx ) = ρ ( x ) } ) > − δ , for all g ∈ S , then we can find y ∈ Y such that µ ( { x ∈ G | ρ ( x ) = y } ) > − ε .Now, let t ∈ G . Using equation 8.1, for all g ∈ Γ and almost every x ∈ G we have ρ t ( gx ) = Θ( gxt )Θ( gx ) − = W ( g, xt )Θ( xt ) ( v ( g, xt ) v ( g, x ) − ) Θ( x ) − W ( g, x ) − . This implies that if v ( g, xt ) = v ( g, x ) and W ( g, xt ) = W ( g, x ), then ρ t ( gx ) = W ( g, x ) ρ t ( x ) W ( g, x ) − .Since v and W take values into countable groups, we have that µ ( { x ∈ G | v ( g, xt ) = v ( g, x ) } ) → µ ( { x ∈ G | W ( g, xt ) = W ( g, x ) } ) →
1, as t → e . Altogether, we conclude that for all g ∈ Γ wehave µ ( { x ∈ G | ρ t ( gx ) = W ( g, x ) ρ t ( x ) W ( g, x ) − } ) →
1, as t → e .This last fact in combination with Claim 1 implies that we can find s, t ∈ G such that the followingfive conditions are satisfied:(1) µ ( { x ∈ G | ρ s ( gx ) = W ( g, x ) ρ s ( x ) W ( g, x ) − } ) > − ε , for all g ∈ F .(2) µ ( { x ∈ G | ρ s ( gx ) = W ( g, x ) ρ s ( x ) W ( g, x ) − } ) > − δ , for all g ∈ S .(3) µ ( { x ∈ G | ρ t ( gx ) = W ( g, x ) ρ t ( x ) W ( g, x ) − } ) > − ε , for all g ∈ F .(4) µ ( { x ∈ G | ρ t ( gx ) = W ( g, x ) ρ t ( x ) W ( g, x ) − } ) > − δ , for all g ∈ S .(5) ( ρ s ( x ) , ρ t ( x )) ∈ A , for almost every x ∈ G .Next, consider the “diagonal conjugation” action ¯ H y ¯ H × ¯ H given by h · ( k, l ) = ( hkh − , hlh − ),for all h ∈ ¯ H and ( k, l ) ∈ ¯ H × ¯ H . Since ¯ H is a real algebraic group, this action is smooth by Theorem2.13. Therefore, the quotient space ¯ Y is a standard Borel space. Analogously, consider the diagonalconjugation action H y H × H and denote by Y the quotient space. Since the left multiplicationaction of Z ( ¯ H ) × Z ( ¯ H ) on ¯ H × ¯ H commutes with the diagonal conjugation action of ¯ H , we havea well-defined left multiplication action Z ( ¯ H ) × Z ( ¯ H ) y ¯ Y . Then Y ≡ ¯ Y / ( Z ( ¯ H ) × Z ( ¯ H )) and weconclude that Y is a standard Borel space. We denote by q : H × H → Y the quotient map. By combining (2) and (4) we get that µ ( { x ∈ G | q ( ρ s ( gx ) , ρ t ( gx )) = q ( ρ s ( x ) , ρ t ( x )) } ) > − δ , for all g ∈ S . Since Y is a standard Borel space we can apply the above consequence of strong ergodicityto the Borel map G ∋ x → q ( ρ s ( x ) , ρ t ( x )) ∈ Y . Thus, we get that there exists y ∈ H × H suchthat µ ( { x ∈ G | ( ρ s ( x ) , ρ t ( x )) ∈ H · y } ) > − ε .This fact and condition (5) imply that we can find x ∈ G such that ( ρ s ( x ) , ρ t ( x )) ∈ A ∩ H · y .Since ( ρ s ( x ) , ρ t ( x )) ∈ A , the stabilizer of ( ρ s ( x ) , ρ t ( x )) in H is trivial. Thus, the stabilizer of y in H is trivial. Hence, we can identify the orbit H · y with H .We define a Borel map α : G → H by letting α ( x ) = ( ( ρ s ( x ) , ρ t ( x )) , if ( ρ s ( x ) , ρ t ( x )) ∈ H · ye, otherwise.Finally, fix g ∈ F . Let C = { x ∈ G | ( ρ s ( gx ) , ρ t ( gx )) = W ( g, x ) · ( ρ s ( x ) , ρ t ( x )) } . The combination ofconditions (1) and (3) yields that µ ( C ) > − ε D = { x ∈ G | α ( x ) = ( ρ s ( x ) , ρ t ( x )) } , then µ ( D ) > − ε > − ε g ∈ F , we get that µ ( g − D ) > − ε µ ( C ∩ D ∩ g − D ) > − ε .If x ∈ C ∩ D ∩ g − D , then we clearly have that α ( gx ) = W ( g, x ) α ( x ), and the claim is proven. (cid:3) Since the action Γ y G is strongly ergodic, combining Claim 2 and Lemma 3.4 implies that wecan find a Borel map α : G → H such that W ( g, x ) = α ( gx ) α ( x ) − , for all g ∈ Γ and almost every x ∈ G . This finishes the first part of the proof. The second part of the proof . Next, we use the conclusion of the first part to derive thetheorem. This part relies on the following claim:
Claim 3.
There exist a Borel map φ : G → Λ, a topological isomorphism δ : G → H , and y ∈ H/ ∆such that δ (Γ) ⊂ Λ and θ ( π ( x )) = φ ( x ) δ ( x ) y , for almost every x ∈ G . Proof of Claim 3 . Denote e Γ = p − (Γ). Since H is a connected real Lie group, we can find a simplyconnected l.c.s.c. group e H together with a continuous onto homomorphisms q : e H → H such thatker( q ) < e H is a discrete subgroup.Let ˜ W : e Γ × e G → Λ and ˜ α : e G → H be given by ˜ W ( g, x ) = W ( p ( g ) , p ( x )) and ˜ α ( x ) = α ( p ( x )).Then ˜ W ( g, x ) = ˜ α ( gx ) ˜ α ( x ) − , for all g ∈ e Γ and almost every x ∈ e G . Since e G is simply connected,by Corollary 3.3, we can find a Borel map φ : e G → Λ and a homomorphism ρ : e Γ → Λ such that(8.2) ˜ W ( g, x ) = φ ( gx ) ρ ( g ) φ ( x ) − , for all g ∈ e Γ and almost every x ∈ e G. Define β : e G → H by letting β ( x ) = φ ( x ) − ˜ α ( x ). Then we get that β ( gx ) = ρ ( g ) β ( x ), for all g ∈ e Γand almost every x ∈ e G . By Lemma 2.14, ρ extends to a continuous homomorphism ρ : e G → H .Our next goal is to show that ρ is onto. Define ˜ θ : e G → H/ ∆ by ˜ θ ( x ) = θ ( π ( p ( x ))). Then˜ θ ( gx ) = ˜ W ( g, x )˜ θ ( x ), for all g ∈ e Γ and almost every x ∈ e G . Thus, if ˆ θ : e G → H/ ∆ is given byˆ θ ( x ) = φ ( x ) − ˜ θ ( x ), then 8.2 implies that(8.3) ˆ θ ( gx ) = ρ ( g )ˆ θ ( x ) , for all g ∈ e Γ and almost every x ∈ e G. Since e Γ < e G is dense and ρ : e G → H is continuos, equation 8.3 holds for every g ∈ e G . By Fubini’stheorem, we can find x ∈ H/ ∆ such that ˆ θ ( gx ) = ρ ( g )ˆ θ ( x ), for almost every g ∈ e G . Thus, if y = ρ ( x ) − ˆ θ ( x ), then ˆ θ ( g ) = ρ ( g ) y , for almost every g ∈ e G . From this we deduce that(8.4) ˜ θ ( x ) = φ ( x ) ρ ( x ) y, for almost every x ∈ e G. RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 27
Since θ : A → B is a nonsingular isomorphism, we get that m H/ ∆ (˜ θ ( e G )) = m H/ ∆ ( θ ( G/ ∆)) > m H ( ρ ( e G )) >
0. This implies that ρ ( e G ) < H is an open subgroup. Since H is connected, we conclude that ρ ( e G ) = H , hence ρ is onto.Next, let us show that ker( ρ ) is a discrete subgroup of e G . To this end, let g n ∈ ker( ρ ) be a sequencewhich converges to the identity. Since φ ( e G ) ⊂ Λ and Λ is countable, there exists N > m e G ( { x ∈ e G | φ ( g n x ) = φ ( x ) and π ( p ( x )) , π ( p ( g n x )) ∈ A } ) >
0, for all n > N . By using 8.4 wefurther get that m e G ( { x ∈ e G | ˜ θ ( g n x ) = ˜ θ ( x ) and π ( p ( x )) , π ( p ( g n x )) ∈ A } ) >
0, for all n > N . Since θ is 1-1 on A , we get that m e G ( { x ∈ e G | p ( g n ) π ( p ( x )) = π ( p ( g n x )) = π ( p ( x )) } ) >
0, for all n > N .Thus, p ( g n ) does not act freely on G/ Σ, hence by Lemma 8.2 we get that p ( g n ) = e , for all n > N .Since ker( p ) is discrete, we deduce that g n = e , for n large enough.Altogether, we have shown that ρ : e G → H is an onto continuous homomorphism with discretekernel, hence e G is a covering group of H . The uniqueness of covering groups implies that we canfind a topological isomorphism τ : e G → e H such that ρ = q ◦ τ . Since G and H have trivial center, weget that ker( p ) = Z ( e G ) and ker( q ) = Z ( e H ) and hence ker( ρ ) = τ − (ker( p )) = τ − ( Z ( e H )) = Z ( e G ).Therefore, ρ descends to a topological isomorphism δ : G → H . Since ρ ( e Γ) ⊂ Λ, we have δ (Γ) ⊂ Λ.Finally, let σ ∈ ker( p ). Then ρ ( σ ) = e and ˜ θ ( xσ ) = ˜ θ ( x ), for all x ∈ e G . By using 8.4 we deducethat φ ( xσ ) ρ ( x ) y = φ ( x ) ρ ( x ) y , for almost every x ∈ e G . Since φ ( x ) , φ ( xσ ) ∈ Λ and Λ is countable,Lemma 8.2 implies that φ ( xσ ) = φ ( x ), for almost every x ∈ e G . Therefore, φ : e G → Λ descends toa map φ : G → Λ. Together with equation 8.4, this proves the claim. (cid:3)
Claim 4. φ : G → Λ factors through the quotient π : G → G/ Σ. Proof of Claim 4.
Since φ : e G → Λ factors through the map p : e G → G by the previous claim,equation 8.2 rewrites as W ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − , for all g ∈ Γ and almost every x ∈ G .Let σ ∈ Σ. Since W factors through the map Γ × G → Γ × G/ Σ, we get that φ ( gxσ ) δ ( g ) φ ( xσ ) − = W ( g, xσ ) = W ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − . Hence, if Φ : G → Λ is given by Φ( x ) = φ ( x ) − φ ( xσ ), then Φ( gx ) = δ ( g )Φ( x ) δ ( g ) − , for all g ∈ Γand almost every x ∈ G . Since δ : G → H is continuous, it follows that Φ( gx ) = δ ( g )Φ( x ) δ ( g ) − ,for all g ∈ G and almost every x ∈ G . By Fubini’s theorem, we can find x ∈ G such thatΦ( gx ) = δ ( g )Φ( x ) δ ( g ) − , for almost every g ∈ G . Thus, if we let k = δ ( x ) − Φ( x ) δ ( x ), thenΦ( g ) = δ ( g ) kδ ( g ) − , for almost every g ∈ G .Since Λ is countable, we can find l ∈ Λ such that C = { g ∈ G | Φ( g ) = l and Φ( g ) = δ ( g ) kδ ( g ) − } satisfies m G ( C ) >
0. Note that k commutes with δ ( C − C ). Since m G ( C ) > δ is onto, we getthat m H ( δ ( C )) > m H ( δ ( C − C )) >
0. Combining the last two facts we derive that thecentralizer of k in H has positive measure, hence is an open subgroup of H . Since H is connectedand has trivial center, we get that k = e . This implies that Φ( x ) = e , for almost every x ∈ G .Thus, φ ( xσ ) = φ ( x ), for almost every x ∈ G . Since σ ∈ Σ is arbitrary, this proves the claim. (cid:3)
Let h ∈ H such that y = h ∆. We end the proof of Theorem E by showing the following: Claim 5. δ (Γ) = Λ and δ (Σ) = h ∆ h − . Proof of Claim 5.
By Claim 3, we have that δ (Γ) ⊂ Λ. To show the reverse inclusion, let g ∈ G such that δ ( g ) ∈ Λ. By Claim 3 we also have that θ ( π ( gx )) = φ ( gx ) δ ( g ) δ ( x ) y ∈ Λ δ ( x ) y = Λ φ ( x ) δ ( x ) y = Λ θ ( π ( x )) , for almost every x ∈ G. Thus, θ ( gx ) ∈ Λ θ ( x ), for almost every x ∈ G/ Σ. Since θ : A → B is a stable orbit equivalence, wededuce that gx ∈ Γ x , for almost every x ∈ G , hence g ∈ Γ. This proves that δ (Γ) = Λ. Next, we show that δ (Σ) ⊂ h ∆ h − . For this, let σ ∈ Σ. By Claim 4 we have that φ ( xσ ) = φ ( x ), foralmost every x ∈ G . By using Claim 3 we get that θ ( π ( xσ )) = φ ( xσ ) δ ( xσ ) h ∆ = φ ( x ) δ ( x ) δ ( σ ) h ∆,for almost every x ∈ G . Since θ ( π ( xσ )) = θ ( π ( x )) = φ ( x ) δ ( x ) h ∆, for almost every x ∈ G , wededuce that δ ( σ ) h ∆ = h ∆. Thus, δ ( σ ) ∈ h ∆ h − , as desired.Finally, we prove that δ − ( h ∆ h − ) ⊂ Σ. To see this, let g ∈ G such that δ ( g ) ∈ h ∆ h − . By usingClaim 3 we get that θ ( π ( xg )) = φ ( xg ) δ ( xg ) h ∆ = φ ( xg ) δ ( x ) h ∆ = φ ( xg ) φ ( x ) − θ ( π ( x )) ∈ Λ θ ( π ( x )),for almost every x ∈ G . Equivalently, we have that θ ( π ( xg )) ∈ Λ θ ( π ( x )), for almost every x ∈ G .Since θ : A → B is an orbit equivalence, we conclude that π ( xg ) ∈ Γ π ( x ), or equivalently, that xg ∈ Γ x Σ, for almost every x ∈ G . Since Γ and Σ are countable, we can find γ ∈ Γ and σ ∈ Σsuch that C = { x ∈ G | xg = γxσ } satisfies m G ( C ) >
0. It is clear that gσ − commutes with C − C .Thus, the centralizer of gσ − in G has positive measure, hence is an open subgroup of G . Since G is connected and has trivial center we conclude that gσ − = e . Hence, g = σ ∈ Σ, as claimed. (cid:3)
This finishes the proof of Theorem 8.1. (cid:4) Proof of Theorem F
In this section we prove Theorem F, in the following more precise form.
Theorem 9.1.
Let ¯ G , ¯ H be connected real algebraic groups and ¯ K < ¯ G , ¯ L < ¯ H be R -subgroupssuch that ∩ g ∈ ¯ G g ¯ Kg − = Z ( ¯ G ) and ∩ h ∈ ¯ H h ¯ Lh − = Z ( ¯ H ) . Denote G = ¯ G/Z ( ¯ G ) , K = ¯ K/Z ( ¯ G ) , H = ¯ H/Z ( ¯ H ) and L = ¯ L/Z ( ¯ H ) . Suppose that K and L are connected.Let Γ < G , Λ < H be countable dense subgroups such that the translation actions Γ y ( G, m G ) and Λ y ( H, m H ) are strongly ergodic.Let A ⊂ G/K and B ⊂ H/L be non-negligible measurable sets and θ : A → B be a nonsingularisomorphism such that θ (Γ x ∩ A ) = Λ θ ( x ) ∩ B , for almost every x ∈ A .Then we can find a Borel map φ : G/K → Λ , a topological isomorphism δ : G → H and h ∈ H such that δ (Γ) = Λ , δ ( K ) = hLh − and θ ( x ) = φ ( x ) δ ( x ) hL , for almost every x ∈ A .Proof. We begin by recording a fact which we will use repeatedly. If g ∈ ¯ G \ Z ( ¯ G ), then Lemma2.11 gives that the set of x ∈ ¯ G such that gx ¯ K = x ¯ K has measure zero. This implies the following: Fact . If g ∈ G \ { e } , then the set of x ∈ G such that gxK = xK has measure zero. Similarly, if h ∈ H \ { e } , then the set of y ∈ H such that hyL = yL has measure zero.As the action Γ y G/K is ergodic, we may extend θ to a countable-to-1 map θ : G/K → H/L such that θ (Γ x ) ⊂ Λ θ ( x ), for almost every x ∈ G/K . Since the action Λ y H/L ergodic, we mayassume that B ⊂ H/L is in fact an open set.The above fact implies that the action Λ y H/L is free and therefore we can define a cocycle w : Γ × G/K → Λ by the formula θ ( gx ) = w ( g, x ) θ ( x ). Let π : G → G/K denote the quotient. Wedefine Θ : G → H/L and W : Γ × G → Λ by Θ( x ) = θ ( π ( x )) and W ( g, x ) = w ( g, π ( x )). Note thatΘ( gx ) = W ( g, x )Θ( x ), for all g ∈ Γ and almost every x ∈ G . The first part of the proof . In the first part of the proof we show that there exists a Borel map α : G → H such that W ( g, x ) = α ( gx ) α ( x ) − , for all g ∈ Γ and almost every x ∈ G . To this end,we follow closely the proof of [Io13, Theorem 4.4].Let µ be a Borel probability measure on G which is equivalent to m G . Claim 1.
There exists a sequence of Borel maps α n : G → H such that for all g ∈ Γ we have that µ ( { x ∈ G | W ( g, x ) = α n ( gx ) α n ( x ) − } ) →
1, as n → ∞ . RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 29
Proof of Claim 1.
Let ε > F ⊂ Γ be finite. Since ¯
L < ¯ H are real algebraic groups and ∩ h ∈ ¯ H h ¯ Lh − = Z ( ¯ H ), Lemma 2.11 provides m >
1, an ¯ H -invariant open conull set Ω ⊂ ( ¯ H / ¯ L ) m and a Borel map τ : Ω → H = ¯ H/Z ( ¯ H ) such that τ ( hx ) = hτ ( x ), for all h ∈ ¯ H and x ∈ Ω. Thus,if we identify ¯ H/ ¯ L with H/L , then we can view Ω as an H -invariant open conull subset of ( H/L ) m which admits an H -equivariant Borel map τ : Ω → H .Since W takes values into a countable group, there exists a neighborhood V of e ∈ G such that(9.1) µ ( { x ∈ G | W ( g, xt ) = W ( g, x ) } ) > − m − ε, for all g ∈ F and every t ∈ V . Now, since θ : G/K → H/L is a nonsingular map and Ω ⊂ ( H/L ) m is conull, we derive that theset { ( x , .., x m ) ∈ ( G/K ) m | ( θ ( x ) , ..., θ ( x m )) ∈ Ω } is conull in ( G/K ) m . Equivalently, we get thatthe set { ( x , ..., x m ) ∈ G m | (Θ( x ) , ..., Θ( x m )) ∈ Ω } is conull in G m . Fubini’s theorem implies thatfor almost every ( t , ..., t m ) ∈ G m , the set { x ∈ G | (Θ( xt ) , ..., Θ( xt m )) ∈ Ω } is conull in G .In particular, we can find t , ..., t m ∈ V such that the set { x ∈ G | (Θ( xt ) , ..., Θ( xt m )) ∈ Ω } isconull in G . We define ψ : G → ( H/L ) m by letting ψ ( x ) = (Θ( xt ) , ..., Θ( xt m )). Further, we define C to be the set of x ∈ G satisfying the following three conditions: • ψ ( x ) ∈ Ω and ψ ( gx ) ∈ Ω, for all g ∈ F , • Θ( gxt i ) = w ( g, xt i )Θ( xt i ), for all g ∈ F and 1 i m , and • W ( g, x ) = W ( g, xt i ), for all g ∈ F and 1 i m .Since ψ ( x ) ∈ Ω and Θ( gx ) = W ( g, x )Θ( x ), for all g ∈ Γ and almost every x ∈ G , equation 9.1(applied to t , ..., t m ∈ V ) implies that µ ( C ) > − ε .Finally, we define α : G → H by letting α ( x ) = ( τ ( ψ ( x )) , if ψ ( x ) ∈ Ω e, if ψ ( x ) Ω . Then for every x ∈ C and g ∈ F we have that α ( gx ) = τ ( ψ ( gx )) = τ (Θ( gxt ) , ..., Θ( gxt m )) = τ ( W ( g, xt )Θ( xt ) , ..., W ( g, xt m )Θ( xt m )) = τ ( W ( g, x )Θ( xt ) , ..., W ( g, x )Θ( xt m )) = W ( g, x ) τ (Θ( xt ) , ..., Θ( xt m )) = W ( g, x ) α ( x ) . Thus, α : G → H is a Borel map which satisfies µ ( { x ∈ G | α ( gx ) = W ( g, x ) α ( x ) } ) > − ε , for all g ∈ F . Since ε > F ⊂ Γ are arbitrary, the claim follows. (cid:3)
Since the action Γ y G is strongly ergodic, Claim 1 and Lemma 3.4 imply that we can find a Borelmap α : G → H such that W ( g, x ) = α ( gx ) α ( x ) − , for all g ∈ Γ and almost every x ∈ G . The second part of the proof . In the second part of proof we obtain the conclusion by using astrategy similar to the one employed in the second part of the proof of Theorem 8.1.We start by adapting part of the proof of Theorem 8.1 to our present context. Since
G, H areconnected real algebraic groups, we can find simply connected l.c.s.c. groups e G , e H together withcontinuous onto homomorphisms p : e G → G , q : e H → H such that ker( p ) < e G , ker( q ) < e H arediscrete subgroups. Denote e Γ = p − (Γ).Let ˜ W : e Γ × e G → Λ and ˜ α : e G → Λ be given by ˜ W ( g, x ) = W ( p ( g ) , p ( x )) and ˜ α ( x ) = α ( p ( x )).Then ˜ W ( g, x ) = ˜ α ( gx ) ˜ α ( x ) − , for all g ∈ e Γ and almost every x ∈ e G . Since e G is simply connected,Corollary 3.3 yields a Borel map φ : e G → Λ and a homomorphism ρ : e Γ → Λ such that(9.2) ˜ W ( g, x ) = φ ( gx ) ρ ( g ) φ ( x ) − , for all g ∈ e Γ and almost every x ∈ e G. Define β : e G → H by letting β ( x ) = φ ( x ) − ˜ α ( x ). Then we get that β ( gx ) = ρ ( g ) β ( x ), for all g ∈ ˜Γand almost every x ∈ ˜ G . By Lemma 2.14, ρ extends to a continuous homomorphism ρ : e G → H . Define ˜ θ : e G → H/L by letting ˜ θ ( x ) = θ ( π ( p ( x ))). By using Fubini’s theorem as in the proof ofClaim 3 in the proof of Theorem 8.1, we find z ∈ H/L such that(9.3) ˜ θ ( x ) = φ ( x ) ρ ( x ) z, for almost every x ∈ G .We are now ready to prove the following claim, which is the core of the second part of the proof: Claim 2 . ρ is onto, ρ ( e Γ) = Λ, and ker( ρ ) < e G is discrete. Proof of Claim 2 . Since ρ is continuous and φ takes countably many values, equation 9.2 impliesthat lim g ∈ e Γ ,g → e ˜ W ( g, x ) = e , for almost every x ∈ e G . This further implies that lim g ∈ Γ ,g → e w ( g, x ) = e ,for almost every x ∈ G/K . Recall that θ : A → B satisfies θ (Γ x ∩ A ) = Λ θ ( x ) ∩ B , for almostevery x ∈ A . Thus, if x ∈ A , then since B is open we get that hθ ( x ) ∈ B , for all h ∈ Λ which areclose enough to e ∈ H . Thus, for all h ∈ Λ close enough to e ∈ H there is v ( h, x ) ∈ Λ such that hθ ( x ) = θ ( v ( h, x ) x ). By repeating the above, with the roles of the two actions reversed (note thatthe assumptions are symmetric), we get that lim h ∈ Λ ,h → e v ( h, x ) = e , for almost every x ∈ A .Let D ⊂ e G be a non-negligible compact set such that D := π ( p ( D )) ⊂ A and φ is constant on D .Let λ ∈ Λ such that φ ( y ) = λ , for all y ∈ D . Note that if y ∈ D , then ˜ θ ( y ) = θ ( π ( p ( y ))) ∈ θ ( A ) = B .Since B is open, it follows that lim h ∈ H,h → e m e G ( { y ∈ D | h ˜ θ ( y ) ∈ B } ) = m e G ( D ).Let y ∈ D and h ∈ Λ such that h ˜ θ ( y ) ∈ B . Denote x = π ( p ( y )). Since hθ ( x ) = h ˜ θ ( y ) ∈ B , by thefirst paragraph there exists v ( h, x ) ∈ Γ such that hθ ( x ) = θ ( v ( h, x ) x ). Let ω ( h, y ) ∈ e Γ such that p ( ω ( h, y )) = v ( h, x ). Since lim h ∈ Λ ,h → e v ( h, x ) = e and ker( p ) < e G is discrete, we may choose ω ( h, y )such that lim h ∈ Λ ,h → e ω ( h, y ) = e . Then h ˜ θ ( y ) = hθ ( x ) = θ ( v ( h, x ) x ) = ˜ θ ( ω ( h, y ) y ) and 9.3 gives that hφ ( y ) ρ ( y ) z = h ˜ θ ( y ) = ˜ θ ( ω ( h, y ) y ) = φ ( ω ( h, y ) y ) ρ ( ω ( h, y ) y ) z = φ ( ω ( h, y ) y ) ρ ( ω ( h, y )) ρ ( y ) z. Since hφ ( y ) , φ ( ω ( h, y ) y ) ρ ( ω ( h, y )) ∈ Λ, Λ is countable and acts freely on
H/L , and ρ ( D ) z ⊂ H/L is non-negligible, we conclude that hφ ( y ) = φ ( ω ( h, y ) y ) ρ ( ω ( h, y )), whenever h ˜ θ ( y ) ∈ B .Next, since lim h ∈ H,h → e m e G ( { y ∈ D | h ˜ θ ( y ) ∈ B } ) = m e G ( D ) and lim h ∈ Λ ,h → e ω ( h, y ) = e , we derive thatlim h ∈ Λ ,h → e m e G ( { y ∈ D | h ˜ θ ( y ) ∈ B and ω ( h, y ) y ∈ D } ) = m e G ( D ) . In particular, we can find a neighborhood V of e ∈ H such that for every h ∈ Λ ∩ V , there is y ∈ D such that h ˜ θ ( y ) ∈ B and ω ( h, y ) y ∈ D . Since φ ( y ) = φ ( ω ( h, y )) = λ (as φ ≡ λ on D ), we get that hλ = λρ ( ω ( h, y )) and ω ( h, y ) ∈ Dy − ⊂ DD − . Hence h ∈ λρ ( e Γ ∩ DD − ) λ − . Since h ∈ Λ ∩ V isarbitrary, we deduce that λ − (Λ ∩ V ) λ ⊂ ρ ( e Γ ∩ DD − ).Since Λ < H is dense and H is connected, Λ ∩ V generates Λ. We deduce that Λ = λ − Λ λ ⊂ ρ ( e Γ).Since ρ ( e Γ) ⊂ Λ, we conclude that ρ ( e Γ) = Λ. Moreover, since Λ < H is dense, ρ is continuous, and DD − is compact, we get that λ − V λ ⊂ ρ ( DD − ). Since H is connected, V generates H , and itfollows that ρ is onto.Finally, to see that ker( ρ ) is discrete, let g n ∈ ker( ρ ) be a sequence such that lim n →∞ g n = e . Since φ takes countably many values, we have that lim n →∞ m e G ( { x ∈ D | φ ( g n x ) = φ ( x ) } ) = m e G ( D ). By usingequation 9.3 we get that lim n →∞ m e G ( { x ∈ D | ˜ θ ( g n x ) = ˜ θ ( x ) } ) = m e G ( D ). Since D ⊂ G/H is compactand lim n →∞ p ( g n ) = e , we also have that lim n →∞ m G/K ( { x ∈ D | p ( g n ) x ∈ D ) } ) = m G/K ( D ). Since π and p are nonsingular, we get that lim n →∞ m e G ( { x ∈ D | π ( p ( g n x )) = p ( g n ) π ( p ( x )) ∈ D } ) = m e G ( D ). RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 31
Altogether, the set of x ∈ D satisfying π ( p ( g n x )) ∈ D and ˜ θ ( g n x ) = ˜ θ ( x ) is non-negligible, forlarge enough n . Since θ is 1-1 on D ⊂ A , we get that p ( g n ) π ( p ( x )) = π ( p ( g n x )) = π ( p ( x )), for allsuch x ∈ D . By using the fact from the beginning of the proof, we conclude that p ( g n ) = e , forlarge enough n . Since ker( p ) < e G is discrete, we must have that g n = e , for large enough n . (cid:3) The rest of the proof is divided between the following three claims.
Claim 3 . The map φ : e G → Λ factors through p : e G → G and there exists a topological isomorphism δ : G → H such that δ (Γ) = Λ and θ ( π ( x )) = φ ( x ) δ ( x ) z , for almost every x ∈ G . Proof of Claim 3.
By Claim 2, ρ : e G → H is an onto continuous homomorphism with discretekernel, hence e G is a covering group of H . The uniqueness of covering groups implies that we canfind a topological isomorphism τ : e G → e H such that ρ = q ◦ τ . Since G and H have trivial center, weget that ker( p ) = Z ( e G ) and ker( q ) = Z ( e H ) and hence ker( ρ ) = τ − (ker( p )) = τ − ( Z ( e H )) = Z ( e G ).Therefore, ρ descends to a topological isomorphism δ : G → H . Since ρ ( e Γ) = Λ, we have δ (Γ) = Λ.Let σ ∈ ker( p ). Then ρ ( σ ) = e and ˜ θ ( xσ ) = ˜ θ ( x ), for all x ∈ e G . By using 9.3 we deduce that φ ( xσ ) ρ ( x ) z = φ ( x ) ρ ( x ) z , for almost every x ∈ e G . Since φ ( x ) , φ ( xσ ) ∈ Λ and the action Λ y H/L is free, we get that φ ( xσ ) = φ ( x ), for almost every x ∈ e G . Therefore, φ : e G → Λ descends to a map φ : G → Λ. Together with equation 9.3, this proves the claim. (cid:3)
Claim 4.
The map φ : G → Λ factors through π : G → G/K . Proof of Claim 4.
Since φ factors through p : e G → G by Claim 3, equation 9.2 can be rewritten as W ( g, x ) = φ ( gx ) δ ( g ) φ ( g ) − , for all g ∈ Γ and almost every x ∈ G . Thus, if k ∈ K , then we havethat φ ( gxk ) δ ( g ) φ ( xk ) − = W ( g, xk ) = W ( g, x ) = φ ( gx ) δ ( g ) φ ( x ) − .Hence, if we define λ k ( x ) = φ ( x ) − φ ( xk ), then λ k ( gx ) = δ ( g ) λ ( x ) δ ( g ) − , for all g ∈ Γ and almostevery x ∈ G . This implies that C k = { x ∈ G | λ k ( x ) = e } is Γ-invariant, for every k ∈ K . Since Λ iscountable we can find a neighborhood V of e ∈ G such that m G ( C k ) >
0, for all k ∈ K ∩ V . Sincethe action Γ y G is ergodic, we deduce that C k = G , almost everywhere, for every k ∈ K ∩ V .Thus, the set K of k ∈ K such that C k = G , almost everywhere, is an open subgroup of K . Since K is connected, we conclude that K = K . This clearly implies the claim. (cid:3) Let h ∈ H such that z = hL . Claim 5 . δ ( K ) = hLh − . Proof of Claim 5.
Let k ∈ K . Then by Claim 4, for almost every x ∈ G we have φ ( xk ) = φ ( x ),hence φ ( x ) δ ( x ) yL = θ ( x ) = θ ( xk ) = φ ( x ) δ ( xk ) yL . Thus, δ ( k ) yL = yL and therefore δ ( k ) ∈ yLy − .To show the reverse inclusion, let g n ∈ δ − ( hLh − ) be a sequence such that lim n →∞ g n = e . Weclaim that g n ∈ K , for n large enough. Indeed, the set of x ∈ π − ( A ) such that φ ( g n x ) = φ ( x )and g n x ∈ π − ( A ) is non-negligible, for large enough n . Since δ ( g n ) yL = yL , by Claim 3 foralmost every such x we have that θ ( xg n K ) = φ ( xg n ) δ ( x ) δ ( g n ) yL = φ ( x ) δ ( x ) yL = θ ( xK ). Since xK, xg n K ∈ A and θ is 1-1 on A , we get that xg n K = xK and hence g n ∈ K .The previous paragraph implies that there exists a neighborhood V of e ∈ G such that we have δ − ( hLh − ) ∩ V ⊂ K . Since δ ( K ) ⊂ hLh − , we derive that K ⊂ δ − ( hLh − ) is an open subgroup.Since L is connected, we get that K = δ − ( hLh − ) and thus δ ( K ) = hLh − . (cid:4) Proofs of Propositions G and H
Proof of Proposition G.
Let α denote the action Γ y ( G , m G ) given by gx = p ( g ) x . (1) In order to show that α is strongly ergodic, by Lemma 2.6 it suffices to show that the inducedaction G y ˜ α ( G/ Γ × G , m G/ Γ × m G ) is strongly ergodic. Note that α is the restriction to Γ of theaction G y ( G , m G ) given by gx = p ( g ) x , for all g ∈ G, x ∈ G . It is well-known (see e.g. [Zi84,Proposition 4.2.22]) that ˜ α is isomorphic to the product action G y β ( G/ Γ × G , m G/ Γ × m G )given by g ( x, y ) = ( gx, p ( g ) y ), for all g ∈ G, x ∈ G/ Γ and y ∈ G .To show that β (and hence ˜ α ) is strongly ergodic, fix a Borel probability measure µ on G whichis equivalent to m G . Then ˜ µ = m G/ Γ × µ is a Borel probability measure on G/ Γ × G which isequivalent to m G/ Γ × m G . Let { A n } be a sequence of measurable subsets of G/ Γ × G satisfying(10.1) lim n →∞ sup g ∈ K ˜ µ ( gA n ∆ A n ) = 0 , for every compact set K ⊂ G. Next, recall that the representation ρ : G → U ( L ( G/ Γ , m G/ Γ ) ⊖ C
1) has spectral gap. Note thatthe restriction of β to G preserves ˜ µ . Let π : G → U ( L ( G/ Γ × G , ˜ µ )) be the associated Koopmanrepresentation. Then π ( G ) acts trivially on the subspace L ( G , µ ) ⊂ L ( G/ Γ × G , ˜ µ ). Moreover,the restriction of π to L ( G/ Γ × G , ˜ µ ) ⊖ L ( G , µ ) is unitarily equivalent to ⊕ ∞ i =1 ρ and thereforehas spectral gap. We denote by P : L ( G/ Γ × G , ˜ µ ) → L ( G , µ ) the orthogonal projection.Finally, equation 10.1 gives that we have sup g ∈ K k π ( g )(1 A n ) − A n k L (˜ µ ) →
0, for every compactset K ⊂ G . By using the spectral gap property described in the previous paragraph we deducethat k A n − P (1 A n ) k L (˜ µ ) →
0. This easily implies that there exists a sequence { B n } of measurablesubsets of G such that ˜ µ ( A n ∆( G/ Γ × B n )) →
0. In combination with 10.1 this further impliesthat sup g ∈ K µ ( gB n ∆ B n ) →
0, for every compact set K ⊂ G . Since the action G y ( G , m G ) isstrongly ergodic by Lemma 2.5, we get that lim n →∞ ˜ µ ( A n )(1 − ˜ µ ( A n )) = lim n →∞ µ ( B n )(1 − µ ( B n )) = 0.(2) Since G has property (T), by [Zi81, Proposition 2.4] the action G y ( G/ Γ , m G/ Γ ) has property(T). Applying [PV08, Proposition 3.5] gives that the action Γ y ( G/G , m G/G ) has property (T).Since the action Γ y ( G/G , m G/G ) is isomorphic to α , we conclude that α has property (T). (cid:4) Proof of Proposition H.
Let ν be a Borel probability measure of X := K \ G which isquasi-invariant under the right G -action. Let π : X → G be a Borel map satisfying Kπ ( x ) = x ,for all x ∈ X . The map θ : K × X → G defined by θ ( k, x ) = kπ ( x ) is a K -equivariant Borelisomorphism (where we consider the action K y K × X given by k ( k ′ x ) = ( kk ′ , x )). We identify G with K × X via θ . Since the push-forward of m K × ν through θ is equivalent to m G , we mayview µ = m K × ν as a probability measure on G which is equivalent to m G .Assume that the action Γ ∩ K y ( K, m K ) has spectral gap. Our goal is to show that the actionΓ y ( G, m G ) is strongly ergodic. Let { A n } be a sequence of measurable subsets of G such thatlim n →∞ µ ( gA n ∆ A n ) = 0, for all g ∈ Γ. For all n and x ∈ X , let A xn = { k ∈ K | ( k, x ) ∈ A n } .Then µ ( gA n ∩ A n ) = Z X m K ( gA xn ∆ A xn ) d ν ( x ), for all n and g ∈ K . Thus, after replacing { A n } witha subsequence, we have that lim n →∞ m K ( gA xn ∆ A xn ) = 0, for all g ∈ Γ ∩ K and almost every x ∈ X .Since the action Γ ∩ K y ( K, m K ) has spectral gap, we get that lim n →∞ m K ( A xn )(1 − m K ( A xn )) = 0,for almost every x ∈ X . Further, this implies that(10.2) lim n →∞ sup g ∈ K µ ( gA n ∆ A n ) = 0 . Next, we need the following result whose proof we postpone until the end of this section.
RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 33
Lemma 10.1.
Let G = SL n ( R ) and K = SO n ( R ) for some n > . Let g ∈ G \ K .Then the group generated by g and K is equal to G . Moreover, if Y ⊂ G is a compact set, then wecan find m > and g , ..., g m ∈ { g, g − } such that Y ⊂ g Kg ...g m − Kg m . Remark 10.2.
Lemma 10.1 implies that if g ∈ G \ K , then there exists N > { g n Kg − n } | n | N generate G . I am grateful to Alireza Salehi-Golsefidy for pointingout to me that the following general result holds: if G is a semisimple algebraic group of real rank r over a local field k , then G is generated by r + 1 compact subgroups.Going back to the proof of Proposition H, let g ∈ Γ \ K and Y ⊂ G be a compact set. By Lemma10.1 we can find g , ..., g m ∈ { g, g − } such that Y ⊂ g Kg ...g m − Kg m . Since g ∈ Γ we havethat lim n →∞ µ ( gA n ∆ A n ) = lim n →∞ µ ( g − A n ∆ A n ) = 0. In combination with equation 10.2, these factsimply that lim n →∞ sup g ∈ Y µ ( gA n ∆ A n ) = 0. Since Y ⊂ G is an arbitrary compact set and the action G y ( G, m G ) is strongly ergodic by Lemma 2.5, we get that lim n →∞ µ ( A n )(1 − µ ( A n )) = 0. Thisshows that the sequence { A n } is trivial. (cid:4) Proof of Lemma 10.1.
We denote by diag( x , ...., x n ) the diagonal n × n matrix whosediagonal entries are x , ..., x n . We also denote by T r : M n ( C ) → C the usual trace.Let H be the group generated by g and K . Let A < G be the subgroup of positive diagonalmatrices. Write g = k dk , where k , k ∈ K and d ∈ A . Write d = diag( λ , ..., λ n ). Since g / ∈ K we have that d = I , hence not all the λ i ’s are equal. We assume for simplicity that λ = λ .Let us first show that H = G in the case n = 2. In this case, d = diag( λ , λ ). For every 0 α × k α := (cid:18) α √ − α −√ − α α (cid:19) ∈ K. A direct computation shows that
T r ( dk α dk ∗ α d ) = α ( λ + λ − λ − λ ) + ( λ + λ ). Let a ∈ A with λ + λ T r ( a ) λ + λ . Then we can find α ∈ [0 ,
1] such that
T r ( a ) = T r ( dk α dk ∗ α d ). Since a and dk α dk ∗ α d are positive 2 × k ∈ K such that a = k ( dk α dk ∗ α d ) k ∗ .Hence a ∈ H . This shows that H ∩ A contains { a ∈ A | λ + λ T r ( a ) λ + λ } . Since λ = λ ,we derive that H ∩ A is an open subgroup of A . Since A is connected we deduce that H ∩ A = A ,therefore A ⊂ H . Since G = KAK and K ⊂ H , we conclude that H = G .Before proceeding to the general case, let us derive an additional fact in the case n = 2. Note thatwe can find a , ..., a ∈ A such that I = a a a a and λ + λ T r ( a i ) λ + λ , for all 1 i k , k , ..., k ∈ K such that I = k dk ...k dk .Now, assume that n > i < j n , we denote by G i,j the subgroup of G consisting of all matrices g = ( g k,l ) ∈ G with the property that g k,l = δ k,l , for all 1 k, l n with { k, l } 6⊂ { i, j } . We clearly have that G i,j ∼ = SL ( R ). We claim that G , ⊂ H .Let g = diag( r λ λ , r λ λ ) ∈ SL ( R ) and view g ∈ G , . By applying the above fact to g , we can find k , k , ..., k ∈ G , ∩ K such that I = k gk ...k gk . Let h = dg − = diag( √ λ λ , √ λ λ , λ , ..., λ n ).Then d = gh and h commutes with G , . It is then immediate that k dk ...k dk = ( k gk ...k gk ) h = h . Since d ∈ H , we get that h ∈ H and further that g = d h − ∈ H . Thus, g ∈ G , ∩ H .Since g K and G , ∩ K ⊂ H , by using the case n = 2, we get that G , ⊂ H .Let 1 i < j n . Then there exists k ∈ K such that G i,j = kG , k − and hence G i,j ⊂ H . Sincethe groups G i,j generate G , we conclude that G = H . To see the moreover assertion, note that since g K , the above gives that g and K generate G . Consequently, we can find p > h , ..., h p ∈ { g, g − } such that A = h Kh ...h p − Kh p isnon-negligible. It follows that A − A contains an open neighborhood V of the identity in G . Since G is connected we get that G = ∪ q > V q . Further, since Y is compact, we can find q > Y ⊂ V q . Therefore, we have that Y ⊂ ( A − A ) q , which proves the claim. (cid:4) Proofs of Corollaries I, J and K
Proof of Corollary I.
Assume that Σ < SL m ( R ) and ∆ < SL n ( R ) are both either(1) discrete subgroups, or(2) connected real algebraic subgroups.Suppose that the actions SL m ( Z [ S − ]) y SL m ( R ) / Σ and SL n ( Z [ T − ]) y SL n ( R ) / ∆ are SOE.Denote by Z m the center of SL m ( R ) and by P SL m ( R ) the quotient group SL m ( R ) /Z m . FollowingExample 1.5, the action SL m ( Z [ S − ]) y SL m ( R ) / Σ is SOE to
P SL m ( Z [ S − ]) y P SL m ( R ) / Σ ,where Σ = (Σ Z m ) /Z m . Moreover, Example 1.8 gives that the action SL m ( Z [ S − ]) y SL m ( R )and hence the action P SL m ( Z [ S − ]) y P SL m ( R ) is strongly ergodic. Similarly, we deduce that SL n ( Z [ T − ]) y SL n ( R ) / ∆ is SOE to P SL n ( Z [ T − ]) y P SL n ( R ) / ∆ , where ∆ = (∆ Z n ) /Z n ,and that the action P SL n ( Z [ T − ]) y P SL n ( R ) is strongly ergodic.Now, in case (1), we have that Σ < P SL m ( R ) and ∆ < P SL n ( R ) are discrete subgroups. In case(2), we have that Σ Z m < SL m ( R ) and ∆ Z n < SL n ( R ) are real algebraic subgroups. Moreover,since Σ and ∆ are connected, we get that Σ ∼ = Σ / (Σ ∩ Z m ) and ∆ ∼ = ∆ / (∆ ∩ Z n ) are connected.Altogether, we can apply Theorem 8.1 in case (1) and Theorem 9.1 in case (2) to conclude that thereexists an isomorphism δ : P SL m ( R ) → P SL n ( R ) such that δ ( P SL m ( Z [ S − ])) = P SL n ( Z [ T − ]).This implies that m = n and there exists g ∈ GL m ( R ) such that either δ ( x ) = gxg − , for all x ∈ P SL m ( R ), or δ ( x ) = g ( x t ) − g − , for all x ∈ P SL m ( R ). It now follows easily that S = T . (cid:4) Proof of Corollary J.
In this subsection, we establish the following more precise versionof Corollary J.
Theorem 11.1.
Let G = SL n ( R ) , for some n > . Let Γ < G be a countable dense subgroup whichcontains the center of G such that the translation action Γ y ( G, m G ) is strongly ergodic.If θ : G → G is a nonsingular isomorphism satisfying θ (Γ x ) = Γ θ ( x ) , for almost every x ∈ G , thenwe can find a Borel map φ : G → Γ , g ∈ GL n ( R ) and h ∈ G such that either θ ( x ) = φ ( x ) gxg − h or θ ( x ) = φ ( x ) g ( x − ) t g − h , for almost every x ∈ G . In particular, θ preserves m G .Proof. Let θ : G → G be a nonsingular isomorphism with θ (Γ x ) = Γ θ ( x ), for almost every x ∈ G . Let Z denote the center of G = SL n ( R ). Put G = G/Z = P SL n ( R ) and Γ = Γ /Z .By Example 1.5, the actions Γ y G and Γ y G are orbit equivalent. Moreover, there existsan orbit equivalence τ : G → G such that τ ( x ) ∈ Γ xZ , for almost every x ∈ G . It follows that τ − ( xZ ) ∈ Γ x , for almost every x ∈ G .Let θ = τ θτ − : G → G . Then θ is a nonsingular isomorphism and θ (Γ x ) = Γ θ ( x ), foralmost every x ∈ G . Since G has trivial center and admits a l.c.s.c. universal covering group, andthe action Γ y G is strongly ergodic, by Theorem 4.1 we can find an automorphism δ : G → G satisfying δ (Γ ) = Γ , a Borel map φ : G → Γ , and h ∈ G such that θ ( x ) = φ ( x ) δ ( x ) h , foralmost every x ∈ G . Thus, there is g ∈ GL n ( R ) such that either δ ( x ) = gxg − , for all x ∈ G , or δ ( x ) = g ( x t ) − g − , for all x ∈ G . In particular, δ lifts to an automorphism to G (defined by the RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 35 same formulas). Since δ (Γ ) = Γ , it follows that δ (Γ) = Γ. Using the properties of τ listed in theprevious paragraph, it is easy to check that θ has the desired form.Moreover, since the conjugation action of GL n ( R ) on G preserves the Haar measure m G , we deducethat θ is measure preserving. (cid:4) Proof of Corollary K.
In this subsection, we prove an OE superrigidity result for actions
P SL m ( Z [ S − ]) y P SL m ( R ) / Σ, where Σ < P SL m ( R ) is an arbitrary discrete subgroup and m > Theorem 11.2.
Let m > be an integer, G ′ = P SL m ( R ) , and Σ ′ < G ′ be a discrete subgroup.Let S be a nonempty set of primes and denote Γ ′ = P SL m ( Z [ S − ]) .Then a free ergodic nonsingular action Λ y ( Y, ν ) of a countable group Λ is SOE to the lefttranslation action Γ ′ y G ′ / Σ ′ if and only we can find a subgroup Λ < Λ , a finite normal subgroup N < Λ , and a normal subgroup M ′ < Σ ′ such that • Λ y Y is induced from some nonsingular action Λ y Y , and • Λ /N y Y /N is conjugate to the left-right multiplication action Γ ′ × Σ ′ /M ′ y G ′ /M ′ givenby ( g, σM ′ ) · xM ′ = gxσ − M ′ , for all g ∈ Γ ′ , σ ∈ Σ ′ and x ∈ G ′ .Proof. Let Γ = SL m ( Z [ S − ]) and G = SL m ( R ). We denote by e G the common universal cover of G and G ′ , and by π : e G → G , π ′ : e G → G ′ the covering homomorphisms. Let e Γ = π − (Γ) = π ′− (Γ ′ ). Claim . There is a subgroup e Γ < e Γ such that g e Γ g − ∩ e Γ is dense in e G , for every g ∈ e Γ, and theaction e Γ y e G has property (T). Proof of the claim.
Fix p ∈ S and denote Γ = SL m ( Z [ 1 p ]). Since m >
3, by Example 1.8, thetranslation action Γ y G has property (T).Moreover, if g ∈ Γ, then g Γ g − ∩ Γ is dense in G . To see this, note first that we can find aninteger N > p ∤ N and N g, N g − ∈ GL m ( Z [ 1 p ]). Consider the quotient homomorphism ρ : Γ → SL m ( Z [ 1 p ] /N Z [ 1 p ]). Then Γ := ker( ρ ) satisfies g − Γ g ⊂ Γ , hence Γ ⊂ g Γ g − ∩ Γ .Since G is connected, Γ < G is dense, and Γ is a finite index subgroup of Γ , we conclude thatΓ is dense in G . This proves our assertion.Let e Γ = π − (Γ ). By Example 1.5 we have that e Γ y e G is SOE to Γ y G and therefore hasproperty (T). Since e G is connected and π is a finite-to-1 map, by using the above assertion, itfollows that g e Γ g − ∩ e Γ is dense in e G , for every g ∈ e Γ. (cid:3) We are now ready to prove the only if assertion of Theorem 11.2. The if assertion follows easily(e.g. by using Example 1.5) and is left to the reader. Let Λ y ( Y, ν ) be a nonsingular action whichis SOE to Γ ′ y G ′ / Σ ′ . Let e Σ = π ′− (Σ) and note that we can identify e G/ e Σ with G ′ / Σ ′ via themap x e Σ π ′ ( x )Σ ′ . Then, under this identification, the actions e Γ y e G/ e Σ and Γ ′ y G ′ / Σ ′ havethe same orbits. We therefore deduce that Λ y Y is SOE to e Γ y e G/ e Σ.Since e G is simply connected, the claim allows us to apply Theorem 6.1. Thus, we can find a normalsubgroup ∆ < e Γ × e Σ, a subgroup Λ < Λ, and a Λ -invariant measurable subset Y ⊂ Y such that • ∆ is discrete in e G × e G , • the left-right multiplication action ∆ y e G admits a measurable fundamental domain, • the left-right multiplication action ( e Γ × e Σ) / ∆ y e G/ ∆ is conjugate to Λ y Y , and • the action Λ y Y is induced from Λ y Y .Next, we claim that ∆ ⊂ Z ( e G ) × e Σ. To see this, let ∆ = { x ∈ e Γ | ( x, e ) ∈ ∆ } . Then ∆ is discretein e G and normal in e Γ. Since e Γ < e G is dense and e G is connected, it follows that ∆ ⊂ Z ( e G ). Now,let ( x, y ) ∈ ∆. Since ∆ is normal in e Γ × e Σ, for every g ∈ e Γ we have that ( gxg − , y ) ∈ ∆, and hence gxg − x − ∈ ∆ . Since ∆ is finite, we derive that gxg − = x , for every g ∈ e Γ that is sufficientlyclose to e . Using again the fact that e Γ < e G is dense and e G is connected, we get that x ∈ Z ( e G ).This shows that ∆ ⊂ Z ( e G ) × e Σ, as claimed.Since Z ( e G ) is finite and ∆ < e Γ × e Σ is normal, we can find a normal subgroup f M < e Σ such that∆ ⊂ Z ( e G ) × f M and the inclusion ∆ ⊂ Z ( e G ) × f M has finite index. We may clearly assume that f M contains Z ( e G ), since the latter group is finite. Then M ′ = π ′ ( f M ) is a normal subgroup of Σ ′ .Moreover, since the kernel of π ′ : e G → G ′ is equal to Z ( e G ), we have that e Γ /Z ( e G ) ∼ = Γ ′ . Also, wehave that e Σ / f M ∼ = Σ ′ /M ′ .Let δ : ( e Γ × e Σ) / ∆ → Λ be the group isomorphism provided by the above conjugacy of actions.Then N := δ (( Z ( e G ) × f M ) / ∆) is a finite normal subgroup of Λ and the action Λ /N y Y /N isconjugate to the left-right multiplication action of [( e Γ × e Σ) / ∆] / [( Z ( e G ) × f M ) / ∆] ∼ = Γ ′ × (Σ ′ /M ′ ) on( e G/ ∆) / [( Z ( e G ) × f M ) / ∆]. Since the latter space can be identified with G ′ /M ′ , we are done. (cid:4) Proof of Corollary K . Let us briefly indicate how Theorem 11.2 implies Corollary K. By applyingTheorem 11.2 in the case Σ ′ = { e } , the first part of Corollary K follows. For the second part ofCorollary K, suppose (in the notation from Theorem 11.2) that Σ ′ < Γ ′ is a lattice. Assume that afree ergodic nonsingular action Λ y Y is SOE to Γ ′ y G ′ / Σ ′ . Then there exist a subgroup Λ < Λand normal subgroups
N < Λ , M ′ < Σ ′ such that the conclusion of Theorem 11.2 holds true.Since Σ ′ < G ′ is a lattice, G ′ has trivial center and R -rank( G ′ ) = m − >
2, Margulis’ normalsubgroup theorem (see e.g. [Zi84, Theorem 8.1.2]) implies that either M ′ = { e } or Σ ′ /M ′ is finite.If M ′ = { e } , then we get that Λ /N y Y /N is conjugate to the left-right multiplication actionΓ ′ × Σ ′ y G ′ . If Σ ′ /M ′ is finite, let δ : Γ ′ × Σ ′ /M ′ → Λ /N be the group homomorphism thatwitnesses the conjugacy between Γ ′ × Σ ′ /M ′ y G ′ /M ′ and Λ /N y Y /N given by Theorem 11.2.Let N ′ < Λ be a finite normal subgroup which contains N and satisfies δ ( { e } × Σ ′ /M ′ ) = N ′ /N .It is now easy to see that the actions Γ ′ y G ′ / Σ ′ and Λ /N ′ y Y /N ′ are conjugate. (cid:4) References [BdS14] Y. Benoist, N. de Saxc´e:
A spectral gap theorem in simple Lie groups , preprint arXiv:1405.1808.[BG02] E. Breuillard, T. Gelander:
On dense free subgroups of Lie groups , J. Algebra (2003), 448-467.[BG04] E. Breuillard, T. Gelander:
A topological Tits alternative , Ann. of Math. (2007), 427-474.[BG06] J. Bourgain, A. Gamburd:
On the spectral gap for finitely-generated subgroups of SU (2), Invent. Math. (2008), 83-121.[CFW81] A. Connes, J. Feldman, B. Weiss: An amenable equivalence relations is generated by a single transformation ,Ergodic Th. Dynam. Sys. (1981), 431-450.[CG85] Y. Carri`ere, E. Ghys: Relations d’´equivalence moyennable sur les groupes de Lie , C. R. Acad. Sc. Paris, t.300, S´erie I, (1985).[CW81] A. Connes, B. Weiss: Property T and asymptotically invariant sequences , Israel J. Math. (1980), 209-210.[FM77] J. Feldman, C.C. Moore, Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras . I, II.Trans. Amer. Math. Soc. (1977), 289-324, 325-359.[Fu99] A. Furman:
Orbit equivalence rigidity , Ann. of Math. (1999), 1083-1108.[Fu09] A. Furman:
A survey of Measured Group Theory , Geometry, Rigidity, and Group Actions, 296-374, TheUniversity of Chicago Press, Chicago and London, 2011.[Ga10] D. Gaboriau:
Orbit equivalence and measured group theory , In Proceedings of the ICM (Hyderabad, India,2010), Vol. III, Hindustan Book Agency, 2010, pp. 1501-1527.
RBIT EQUIVALENCE RIGIDITY FOR TRANSLATION ACTIONS 37 [Ge03] S. Gefter:
Fundamental groups of some ergodic equivalence relations of type II ∞ , Qual. Theory Dyn. Syst. (2003), 115-124.[GMO08] A. Gorodnik, F. Maucourant, H. Oh: Manin’s and Peyre’s conjectures on rational points and adelic mixing ,Ann. Sci. ´Ec. Norm. Sup´er. (4) (2008), no. 3, 383-435.[Io08] A. Ioana: Cocycle superrigidity for profinite actions of property (T) groups , Duke Math. J. (2011),337-367.[Io09] A. Ioana:
Relative property (T) for the subequivalence relations induced by the action of SL ( Z ) on T ,Adv. Math. (2010), 1589-1617.[Io13] A. Ioana: Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap , preprintarXiv:1309.3026.[IS10] A. Ioana, Y. Shalom:
Rigidity for equivalence relations on homogeneous spaces , Groups Geom. Dyn. (2013), 403-417.[Ke95] A. S. Kechris: Classical descriptive set theory , Graduate Texts in Mathematics, 156. Springer-Verlag, NewYork, 1995. xviii+402 pp.[Ki06] Y. Kida:
Measure equivalence rigidity of the mapping class group , Ann. of Math. (2010), 1851-1901.[KM99] D. Kleinbock, G. Margulis:
Logarithm laws for flows on homogeneous spaces , Invent. Math. (1999),451-494.[MZ55] D. Montgomery, L. Zippin:
Topological transformation groups , Interscience Publishers, New York-London,1955. xi+282 pp.[OP07] N. Ozawa, S. Popa:
On a class of II factors with at most one Cartan subalgebra , Ann. of Math. (2010),713-749.[Pi04] M. Pichot: Sur la th´eorie spectrale des relations d’´equivalence mesur´ees , J. Inst. Math. Jussieu (2007),no. 3, 453-500.[Po86] S. Popa: Correspondences
On a class of type II factors with Betti numbers invariants . Ann. of Math. (2006), 809-899.[Po05] S. Popa: Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups , Invent. Math. (2007), 243-295.[Po06] S. Popa:
On the superrigidity of malleable actions with spectral gap , J. Amer. Math. Soc. (2008), 981-1000.[Po07] S. Popa: Deformation and rigidity for group actions and von Neumann algebras , In Proceedings of the ICM(Madrid, 2006), Vol. I, European Mathematical Society Publishing House, 2007, 445-477.[PV08] S. Popa, S. Vaes:
Cocycle and orbit superrigidity for lattices in SL ( n, R ) acting on homogeneous spaces ,Geometry, rigidity, and group actions, 419-451, Chicago Lectures in Math., Univ. Chicago Press, Chicago,IL, 2011.[Sc81] K. Schmidt: Asymptotically invariant sequences and an action of SL (2 , Z ) on the 2-sphere , Israel J. Math. (1980), 193-208.[Zi80] R. Zimmer: Strong rigidity for ergodic actions of semisimple Lie groups , Ann. of Math. (1980), 511-529.[Zi81] R. Zimmer:
On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups , Amer.J. Math. (1981), no. 5, 937-951.[Zi84] R. Zimmer:
Ergodic theory and semisimple groups , Monographs in Mathematics, 81. Birkh¨auser Verlag,Basel, 1984. x+209 pp.[Zi87] R. Zimmer:
Amenable actions and dense subgroups of Lie groups , J. Funct. Anal. (1987), 58-64.[Wi02] J. Winkelmann: Generic subgroups of Lie groups , Topology (2002), no. 1, 163-181. Mathematics Department; University of California, San Diego, CA 90095-1555 (United States).
E-mail address ::