aa r X i v : . [ m a t h . D S ] A ug To Robert Zimmer on the occasion of his 60th birthday..
A SURVEY OF MEASURED GROUP THEORY
ALEX FURMAN
Abstract.
The title refers to the area of research which studies infinite groups usingmeasure-theoretic tools, and studies the restrictions that group structure imposes on er-godic theory of their actions. The paper is a survey of recent developments focused onthe notion of Measure Equivalence between groups, and Orbit Equivalence between groupactions. We discuss known invariants and classification results (rigidity) in both areas.
Contents
1. Introduction 2Disclaimer 2Acknowledgements 3Organization of the paper 32. Preliminary discussion and remarks 42.1. Lattices, and other countable groups 42.2. Orbit Equivalence in Ergodic Theory 72.3. Further comments on QI, ME and related topics 93. Measure Equivalence between groups 113.1. Measure Equivalence Invariants 113.2. Orbit/Measure Equivalence Rigidity 203.3. How many Orbit Structures does a given group have? 233.4. Mostow-type rigidity: ME and QI approaches 254. Ergodic equivalence relations 274.1. Basic definitions 274.2. Invariants of equivalence relations 294.3. Rigidity of equivalence relations 395. Techniques 435.1. Superrigidity in semi-simple Lie groups 435.2. Superrigidity for product groups 455.3. Strong rigidity for cocycles 465.4. Cocycle superrigid actions 465.5. Constructing representations 475.6. Local rigidity for measurable cocycles 485.7. Cohomology of cocycles 495.8. Proofs of some results 52Appendix A. Cocycles 55
Date : August 10, 2010.Supported by NSF Grant DMS 0604611, and BSF Grant 2004345.
A.1. The canonical class of a lattice, (co-)induction 55A.2. OE-cocycles 56A.3. ME-cocycles 56References 571.
Introduction
This survey concerns an area of mathematics which studies infinite countable groupsusing measure-theoretic tools, and studies Ergodic Theory of group actions, emphasizing theimpact of group structure on the actions.
Measured Group Theory is a particularly fittingtitle as it suggests an analogy with
Geometric Group Theory . The origins of MeasuredGroup Theory go back to the seminal paper of Robert Zimmer [119], which established adeep connection between questions on Orbit Equivalence in Ergodic Theory to the theoryof lattices in semi-simple groups, specifically to Margulis’ celebrated superrigidity. Thenotion of amenable actions, introduced by Zimmer in an earlier work [118], became anindispensable tool in the field. Zimmer continued to study orbit structures of actions oflarge groups in [25, 34, 120–124, 126, 127] and [115]. The monograph [126] had a particularlybig impact on both ergodic theorists and people studying big groups, as well as researchersin other areas, such as Operator Algebras and Descriptive Set Theory .In the recent years several new layers of results have been added to what we calledMeasured Group Theory, and this paper aims to give an overview of the current state ofthe subject. Such a goal is unattainable – any survey is doomed to be partial, biased,and outdated before it appears. Nevertheless, we shall try our best, hoping to encouragefurther interest in the subject. The reader is also referred to Gaboriau’s paper [49], whichcontains a very nice overview of some of the topics discussed here, and to Shalom’s survey[113] which is even closer to the present paper (hence the similarity of the titles). Themonographs by Kechris and Miller [71] and the forthcoming one [70] by Kechris includetopics in Descriptive Set Theory related to Measured Group Theory. For topics related tovon Neumann algebra we refer to Vaes’ [116] and Popa’s [97] and references therein.The scope of this paper is restricted to interaction of infinite Groups with Ergodic the-ory, leaving out the connections to the theory of von Neumann algebras and Descriptive SetTheory. When possible, we try to indicate proofs or ideas of proofs for the stated results.In particular, we chose to include a proof of one cocycle superrigidity theorem 5.20, whichenables a self-contained presentation of a number of important results: a very rigid equiv-alence relation (Theorem 4.19) with trivial fundamental group and outer automorphismgroup (Theorem 4.15), an equivalence relation not generated by an essentially free actionof any group ( § Disclaimer.
As usual, the quoted results are often presented not in full generality available,and the reader should consult the original papers for full details. The responsibility forinaccuracies, misquotes and other flaws lies with the author of these notes. Zimmer’s cocycle superrigidity proved in [119] plays a central role in another area of research, vigorouslypursued by Zimmer and other, concerning actions of large groups on manifolds. David Fisher surveys thisdirection in [35] in this volume.
SURVEY OF MEASURED GROUP THEORY 3
Acknowledgements.Organization of the paper.
The paper is organized as follows: the next section is devotedto a general introduction which emphasizes the relations between Measure Equivalence,Quasi-Isometry and Orbit Equivalence in Ergodic Theory. One may choose to skip most ofthis, but read Definition 2.1 and the following remarks. Section 3 concerns groups consideredup to Measure Equivalence. Section 4 focuses on the notion of equivalence relations withorbit relations as a prime (but not only) example. In both of these sections we considerseparately the invariants of the studied objects (groups and relations) and rigidity results,which pertain to classification. Section 5 describes the main techniques used in these theories(mostly for rigidity): a discussion of superrigidity phenomena and some of the ad hoc toolsused in the subject; generalities on cocycles appear in the appendix A.
ALEX FURMAN Preliminary discussion and remarks
This section contains an introduction to Measure Equivalence and related topics andcontains a discussion of this framework. Readers familiar with the subject (especially defi-nition 2.1 and the following remarks) may skip to the next section in the first reading.There are two natural entry points to Measured Group Theory, corresponding to theergodic-theoretic and group-theoretic perspectives. Let us start from the latter.2.1.
Lattices, and other countable groups.
When should two infinite discrete groupsbe viewed as closely related? Isomorphism of abstract groups is an obvious, maybe triv-ial, answer. The next degree of closeness would be commensurability : two groups arecommensurable if they contain isomorphic subgroups of finite index. This relation mightbe relaxed a bit further, by allowing to pass to a quotient modulo finite normal subgroups.The algebraic notion of being commensurable, modulo finite kernels, can be vastly general-ized in two directions: Measure Equivalence (Measured Group Theory) and Quasi-Isometry(Geometric Group Theory).The key notion discussed in tis paper is that of
Measure Equivalence of groups. Itwas introduced by Gromov in [57, 0.5.E].
Definition 2.1.
Two infinite discrete countable groups Γ, Λ are
Measure Equivalent (abbreviated as ME, and denoted Γ ME ∼ Λ) if there exists an infinite measure space (Ω , m )with a measurable, measure preserving action of Γ × Λ, so that the actions of each of thegroups Γ and Λ admit finite measure fundamental domains:Ω = G γ ∈ Γ γY = G λ ∈ Λ λX. The space (Ω , m ) is called a (Γ , Λ)- coupling or ME-coupling. The index of Γ to Λ in Ω isthe ratio of the measures of the fundamental domains[Γ : Λ] Ω = m ( X ) m ( Y ) (cid:18) = meas(Ω / Λ)meas(Ω / Γ) (cid:19) . We shall motivate this definition after making a few immediate comments. (a)
The index [Γ : Λ] Ω is well defined – it does not depend on the choice of the fundamentaldomains X , Y for Ω / Λ, Ω / Γ respectively, because their measures are determined by thegroup actions on (Ω , m ). However, a given pair (Γ , Λ) might have ME-couplings withdifferent indices (the set { [Γ : Λ] Ω } is a coset of a subgroup of R ∗ + corresponding to indices[Γ : Γ] Ω of self Γ-couplings. Here it makes sense to focus on ergodic couplings only). (b) Any ME-coupling can be decomposed into an integral over a probability space of ergodic
ME-couplings, i.e., ones for which the Γ × Λ-action is ergodic. (c)
Measure Equivalence is indeed an equivalence relation between groups: for any count-able Γ the action of Γ × Γ on Ω = Γ with the counting measure m Γ by ( γ , γ ) : γ γ γγ −
12 SURVEY OF MEASURED GROUP THEORY 5 provides the trivial self ME-coupling, giving reflexivity; symmetry is obvious from the defi-nition ; while transitivity follows from the following construction of composition of fusion of ME-couplings. If (Ω , m ) is a (Γ , Γ ) coupling and (Ω ′ , m ′ ) is a (Γ , Γ ) coupling, thenthe quotient Ω ′′ = Ω × Γ Ω ′ of Ω × Ω ′ under the diagonal action γ : ( ω, ω ′ ) ( γ ω, γ − ω ′ )inherits a measure m ′′ = m × Γ m ′ so that (Ω ′′ , m ′′ ) becomes a (Γ , Γ ) coupling structure.The indices satisfy: [Γ : Γ ] Ω ′′ = [Γ : Γ ] Ω · [Γ : Γ ] Ω ′ . (d) The notion of ME can be extended to the broader class of all unimodular locallycompact second countable groups: a ME-coupling of G and H is a measure space (Ω , m )with measure space isomorphisms i : ( G, m G ) × ( Y, ν ) ∼ = (Ω , m ) , j : ( H, m H ) × ( X, µ ) ∼ = (Ω , m )with ( X, µ ), (
Y, ν ) being finite measure spaces, so that the actions G y (Ω , m ), H y (Ω , m )given by g : i ( g ′ , y ) i ( gg ′ , y ), h : j ( h ′ , x ) j ( hh ′ , x ), commute. The index is defined by[ G : H ] Ω = µ ( X ) /ν ( Y ). (e) Measure Equivalence between countable groups can be viewed as a category , whose objects are countable groups and morphisms between, say Γ and Λ, are possible (Γ , Λ)couplings. Composition of morphisms is the operation of composition of ME-couplings asin (c). The trivial ME-coupling (Γ , m Γ ) is nothing but the identity of the object Γ. It is alsouseful to consider quotient maps Φ : (Ω , m ) → (Ω , m ) between (Γ , Λ)-couplings (theseare 2-morphisms in the category), which are assumed to be Γ × Λ non-singular maps, i.e.,Φ ∗ [ m ] ∼ m . Since preimage of a fundamental domain is a fundamental domain, it follows(under ergodicity assumption) that m (Φ − ( E )) = c · m ( E ), E ⊂ Ω , where 0 < c < ∞ .ME self couplings of Γ which have the trivial Γ-coupling are especially useful, their cocyclesare conjugate to isomorphisms. Similarly, (Γ , Λ)-couplings which have a discrete couplingas a quotient, correspond to virtual isomorphisms (see Lemma 4.18). (f )
Finally, one might relax the definition of quotients by considering equivariant mapsΦ : Ω → Ω between (Γ i , Λ i )-couplings (Ω i , m i ) with respect to homomorphisms Γ → Γ ,Λ → Λ with finite kernels and co-kernels.Gromov’s motivation for ME comes from the theory of lattices. Recall that a subgroup Γof a locally compact second countable ( lcsc for short) group G is a lattice if Γ is discrete in G and the quotient space G/ Γ carries a finite G -invariant Borel regular measure (necessarilyunique up to normalization); equivalently, if the Γ-action on G by left (equivalently, right)translations admits a Borel fundamental domain of finite positive Haar measure. A discretesubgroup Γ < G with G/ Γ being compact is automatically a lattice. Such lattices are called uniform or cocompact ; others are non-uniform . The standard example of a non-uniformlattice is Γ = SL n ( Z ) in G = SL n ( R ). Recall that a lcsc group which admits a lattice isnecessarily unimodular.II -relA common theme in the study of lattices (say in Lie, or algebraic groups over local fields)is that certain properties of the ambient group are inherited by its lattices. From this One should formally distinguish between (Ω , m ) as a (Γ , Λ) coupling, and the same space with the sameactions as a (Λ , Γ) coupling; hereafter we shall denote the latter by ( ˇΩ , ˇ m ). Example 2.2 illustrates the needto do so. ALEX FURMAN perspective it is desirable to have a general framework in which lattices in the same groupare considered equivalent. Measure Equivalence provides such a framework.
Example 2.2.
If Γ and Λ are lattices in the same lcsc group G , then Γ ME ∼ Λ; the group G with the Haar measure m G is a (Γ , Λ) coupling where( γ, λ ) : g γgλ − . (In fact, Γ ME ∼ G ME ∼ Λ if ME is considered in the broader context of unimodular lcsc groups: G × { pt } ∼ = Γ × G/ Γ). This example also illustrates the fact that the dual (Λ , Γ)-couplingˇ G is better related to the original (Γ , Λ)-coupling G via g g − rather than the identitymap.In Geometric Group Theory the basic notion of equivalence is quasi-isometry (QI).Two metric spaces ( X i , d i ), i = 1 , X QI ∼ X ) if there existmaps f : X → X , g : X → X , and constants M, A so that d ( f ( x ) , f ( x ′ )) < M · d ( x, x ′ ) + A ( x, x ′ ∈ X ) d ( g ( y ) , g ( y ′ )) < M · d ( y, y ′ ) + A ( y, y ′ ∈ X ) d ( g ◦ f ( x ) , x ) < A ( x ∈ X ) d ( f ◦ g ( y ) , y ) < A ( y ∈ X ) . Two finitely generated groups are QI if their Cayley graphs (with respect to some/any finitesets of generators) are QI as metric spaces. It is easy to see that finitely generated groupscommensurable modulo finite groups are QI.Gromov observes that QI between groups can be characterized as
Topological Equiv-alence (TE) defined in the the following statement.
Theorem 2.3 (Gromov [57, Theorem 0.2.C ′ ]) . Two finitely generated groups Γ and Λ arequasi-isometric iff there exists a locally compact space Σ with a continuous action of Γ × Λ ,where both actions Γ y Σ and Λ y Σ are properly discontinuous and cocompact. The space X in the above statement is called a TE-coupling . Here is an idea for theproof. Given a TE-coupling Σ one obtains a quasi-isometry from any point p ∈ Σ bychoosing f : Γ → Λ, g : Λ → Γ so that γp ∈ f ( γ ) X and λp ∈ g ( λ ) Y , where X, Y ⊂ Σ areopen sets with compact closures and Σ = S γ ∈ Γ γY = S λ ∈ Λ λX . To construct a TE-couplingΣ from a quasi-isometry f : Γ → Λ, consider the pointwise closure of the Γ × Λ-orbit of f in the space of all maps Γ → Λ where Γ acts by pre-composition on the domain and Λ bypost-composition on the image. The details are left to the reader.A nice instance of QI between groups is a situation where the groups admit a common geometric model . Here a geometric model for a finitely generated group Γ is a (com-plete) separable metric space (
X, d ) with a properly discontinuous and cocompact action ofΓ on X by isometries . If X is a common geometric model for Γ and Γ , then Γ QI ∼ X QI ∼ Γ .For example, fundamental groups Γ i = π ( M i ) of compact locally symmetric manifolds M and M with the same universal cover ˜ M ∼ = ˜ M = X have X as a common geometricmodel. Notice that the common geometric model X itself does not serve as a TE-coupling SURVEY OF MEASURED GROUP THEORY 7 because the actions of the two groups do not commute. However, a TE-coupling can beexplicitly constructed from the group G = Isom( X, d ), which is locally compact (in fact,compactly generated due to finite generation assumption on Γ i ) second countable group.Indeed, the isometric actions Γ i y ( X, d ) define homomorphisms Γ i → G with finite kernelsand images being uniform lattices. Moreover, the converse is also true: if Γ , Γ admit ho-momorphisms with finite kernels and images being uniform lattices in the same compactlygenerated second countable group G , then they have a common geometric model – take G with a (pseudo-)metric arising from an analogue of a word metric using compact sets.Hence all uniform lattices in the same group G are QI to each other. Yet, typically, non-uniform lattices in G are not QI to uniform ones – see Farb’s survey [30] for the completepicture of QI for lattices in semi-simple Lie groups. To summarize this discussion : the notion of Measure Equivalence is an equivalence re-lation between countable groups, an important instance of which is given by groups whichcan imbedded as lattices (uniform or not) in the same lcsc group. It can be viewed as ameasure-theoretic analogue of the equivalence relation of being Quasi-Isometric (for finitelygenerated groups), by taking Gromov’s Topological Equivalence point of view. An impor-tant instance of QI/TE is given by groups which can be imbedded as uniform lattices in thesame lcsc group. In this situation one has both ME and QI. However, we should emphasizethat this is merely an analogy : the notions of QI and ME do not imply each other.2.2.
Orbit Equivalence in Ergodic Theory.
Ergodic Theory investigates dynamicalsystems from measure-theoretic point of view. The standard setup would include a group Γof measurable transformations of a standard non-atomic measure space ( X, X , µ ). Here weshall focus on actions of discrete countable groups, acting by measure preserving transfor-mations on a standard probability space ( X, µ ) (hereafter probability measure preserving, or p.m.p. actions). It is often convenient to assume the action to be ergodic , i.e., to requireall measurable Γ-invariant sets to be null or co-null (that is µ ( E ) = 0 or µ ( X \ E ) = 0).The classical theory is concerned with actions of Γ = Z , but here we shall focus on largergroups. Focussing on the measure-theoretic structure, everything is considered modulo nullsets.A basic question in this concerns the orbit structure of the action, captured by thefollowing notions of Orbit Equivalence. Definition 2.4.
Two actions Γ y ( X, µ ) and Λ y ( Y, ν ) are orbit equivalent (abbrevi-ated OE, denoted Γ y ( X, µ ) OE ∼ Λ y ( Y, ν )) if there exists a measure space isomorphism T : ( X, µ ) ∼ = ( Y, ν ) which takes Γ-orbits onto Λ-orbits. More precisely, an orbit equivalenceis a Borel isomorphism T : X ′ ∼ = Y ′ between co-null subsets X ′ ⊂ X and Y ′ ⊂ Y with T ∗ µ ( E ) = µ ( T − E ) = ν ( E ), E ⊂ Y ′ and T (Γ .x ∩ X ′ ) = Λ .T ( x ) ∩ Y ′ for x ∈ X ′ .A weak OE , or Stable OE (SOE) is a Borel isomorphism T : X ′ ∼ = Y ′ between positivemeasure subsets X ′ ⊂ X and Y ′ ⊂ Y with T ∗ µ X ′ = ν Y ′ , where µ X ′ = µ ( X ′ ) − · µ | X ′ , ν Y ′ = ν ( Y ′ ) − · ν | Y ′ , so that T (Γ .x ∩ X ′ ) = Λ .T ( x ) ∩ Y ′ for all x ∈ X ′ . The index of suchSOE-map T is µ ( Y ′ ) /ν ( X ′ ).In the study of orbit structure of dynamical systems in the topological or smooth categoryone often looks at such concepts as fixed or periodic points/orbits. Despite the important ALEX FURMAN role these notions play in the underlying dynamical system, they are typically invisible fromthe purely measure-theoretic standpoint being null sets. Hence OE in Ergodic Theory is astudy of the orbit structure as a whole. This point of view is consistent with the generalphilosophy of ”non-commutative measure theory”, i.e. von Neumann algebras. SpecificallyOE in Ergodic Theory is closely related to the theory of II factors as follows.In the 1940s Murray and von Neumann introduced the so called ”group-measure space”construction to provide interesting examples of von-Neumann factors : given a probabilitymeasure preserving (or more generally, non-singular) group action Γ y ( X, µ ) the associatedvon-Neumann algebra M Γ y X is a cross-product of Γ with the Abelian algebra L ∞ ( X, µ ),namely the weak closure in bounded operators on L (Γ × X ) of the algebra generated by theoperators { f ( g, x ) f ( γg, γ.x ) : γ ∈ Γ } and { f ( g, x ) φ ( x ) f ( g, x ) : φ ∈ L ∞ ( X, µ ) } . Er-godicity of Γ y ( X, µ ) is equivalent to M Γ y X being a factor. It turns out that (for essentially free ) OE actions Γ y X OE ∼ Λ y Y the associated algebras are isomor-phic M Γ y X ∼ = M Λ y Y , with the isomorphism identifying the Abelian subalgebras L ∞ ( X )and L ∞ ( Y ). The converse is also true (one has to specify, in addition, an element in H (Γ y X, T )) – see Feldman-Moore [32, 33]. So Orbit Equivalence of (essentially freep.m.p. group actions) fits into the study of II factors M Γ y X with a special focus on theso called Cartan subalgebra given by L ∞ ( X, µ ). We refer the reader to Popa’s 2006 ICMlecture [97] and Vaes’ Seminar Bourbaki paper [116] for some more recent reports on thisrapidly developing area.The above mentioned assumption of essential freeness of an action Γ y ( X, µ ) meansthat, up to a null set, the action is free; equivalently, for µ -a.e. x ∈ X the stabilizer { γ ∈ Γ : γ.x = x } is trivial. This is a natural assumption, when one wants the actinggroup Γ to ”fully reveal itself” in a.e. orbit of the action. Let us now link the notions ofOE and ME. Theorem 2.5.
Two countable groups Γ and Λ are Measure Equivalent iff they admit es-sentially free (ergodic) probability measure preserving actions Γ y ( X, µ ) and Λ y ( Y, ν ) which are Stably Orbit Equivalent. (SOE) = ⇒ (ME) direction is more transparent in the special case of Orbit Equivalence,i.e., index one. Let α : Γ × X → Λ be the cocycle associated to an orbit equivalence T : ( X, µ ) → ( Y, ν ) defined by T ( g.x ) = α ( g, x ) .T ( x ) (here freeness of Λ y Y is used).Consider (Ω , m ) = ( X × Λ , µ × m Λ ) with the actions(2.1) g : ( x, h ) ( gx, α ( g, x ) h ) , h : ( x, k ) ( x, hk − ) ( g ∈ Γ , h ∈ Λ) . Then X × { } is a common fundamental domain for both actions (note that here freenessof Γ y X is used). Of course, the same coupling (Ω , m ) can be viewed as ( Y × Γ , ν × m Γ )with the Λ-action defined using β : Λ × Y → Γ given by T − ( h.y ) = β ( h, y ) .T − ( y ). In themore general setting of Stable
OE one needs to adjust the definition for the cocycles (see[37]) to carry out a similar construction.Alternative packaging for the (OE) = ⇒ (ME) argument uses the language of equivalencerelations (see § Y with X via T − , one views R Λ y Y and R Γ y X as a single von Neumann algebras with center consisting only of scalars. SURVEY OF MEASURED GROUP THEORY 9 relation R . Taking Ω = R equipped with the measure ˜ µ consider the actions g : ( x, y ) ( g.x, y ) , h : ( x, y ) ( x, h.y ) ( g ∈ Γ , h ∈ Λ) . Here the diagonal embedding X R , x ( x, x ), gives the fundamental domain for bothactions.(ME) = ⇒ (SOE). Given an ergodic (Γ , Λ) coupling (Ω , m ), let
X, Y ⊂ Ω be fundamentaldomains for the Λ, Γ actions; these may be chosen so that m ( X ∩ Y ) >
0. The finitemeasure preserving actions(2.2) Γ y X ∼ = Ω / Λ , Λ y Y ∼ = Ω / Γ . have weakly isomorphic orbit relations, since they appear as the restrictions of the II ∞ relation R Γ × Λ y Ω to X and Y and coincide on X ∩ Y . The index of this SOE coincides withthe ME-index [Γ : Λ] Ω (if [Γ : Λ] Ω = 1 one can find a common fundamental domain X = Y ).The only remaining issue is that the actions Γ y X ∼ = Ω / Λ, Λ y Y ∼ = Ω / Γ may not beessential free. This can be fixed (see [47]) by passing to an extension Φ : ( ¯Ω , ¯ m ) → (Ω , m )where Γ y ¯Ω / Λ and Λ y ¯Ω / Γ are essentially free. Indeed, take ¯Ω = Ω × Z × W , whereΛ y Z and Λ y W are free probability measure preserving actions and let g : ( ω, z, w ) ( gω, gz, w ) , h : ( ω, z, w ) ( hω, z, hw ) ( g ∈ Γ , h ∈ Λ) . Remark 2.6.
Freeness of actions is mostly used in order to define the rearrangementcocycles for a (stable) orbit equivalence between actions. However, if SOE comes from aME-coupling the well defined ME-cocycles satisfy the desired rearrangement property (suchas T ( g.x ) = α ( g, x ) .T ( x )) and freeness becomes superfluous.If Φ : ¯Ω → Ω is as above, and ¯ X, ¯ Y denote the preimages of X, Y , then ¯ X, ¯ Y are Λ , Γfundamental domains, the OE-cocycles Γ y ¯ X SOE ∼ Λ y ¯ Y coincide with the ME-cocyclesassociated with X, Y ⊂ Ω.Another, essentially equivalent, point of view is that ME-coupling defines a weak iso-morphism between the groupoids Γ y Ω / Λ and Λ y Ω / Γ. In case of free actions thesegroupoids reduce to their relations groupoids , but in general the information about stabi-lizers is carried by the ME-cocycles.2.3.
Further comments on QI, ME and related topics.
Let Σ be Gromov’s Topo-logical Equivalence between Γ and Λ. Then any point x ∈ Σ defines a quasi-isometry q x : Γ → Λ (see the sketch of proof of Theorem 2.3). In ME the values of the ME-cocycles α ( − , x ) : Γ → Λ play a similar role, however due to their measure-theoretic nature noneof them exists individually, but only as a measurable family. This measurable family isequipped with a probability measure which is invariant under a certain Γ-action.2.3.1.
Using ME for QI.
Although Measure Equivalence and Quasi Isometry are parallel inmany ways, these concepts are different and neither implies the other. Yet, Yehuda Shalomhas shown [112] how one can use ME ideas to study QI of amenable groups. The basicobservation is that a topological coupling Σ of amenable groups Γ and Λ carries a Γ × Λ-invariant measure m (coming from a Γ-invariant probability measure on Σ / Λ), which givesa measure equivalence. It can be thought of as an invariant distribution on quasi-isometriesΓ → Λ, and can be used to induce unitary representations and cohomology with unitarycoefficients etc. from Λ to Γ. Using such constructions, Shalom [112] was able to obtain a list of new QI invariants in the class of amenable groups, such as (co)-homology over Q ,ordinary Betti numbers β i (Γ) among nilpotent groups etc. Shalom also studied the notionof uniform embedding (UE) between groups and obtained group invariants which are monotonic with respect to UE.In [105] Roman Sauer obtains further QI-invariants and UE-monotonic invariants usinga combination of QI, ME and homological methods.In another work [106] Sauer used ME point of view to attack problems of purely topo-logical nature, related to the work of Gromov.2.3.2. ℓ p -Measure Equivalence. Let Γ and Λ be finitely generated groups, equipped withsome word metrics | · | Γ , | · | Λ . We say that a (Γ , Λ) coupling (Ω , m ) is ℓ p for some 1 ≤ p ≤ ∞ if there exist fundamental domains X, Y ⊂ Ω so that the associated ME-cocycles (see A.3) α : Γ × X → Λ and β : Λ × Y → Γ satisfy ∀ g ∈ Γ : | α ( g, − ) | Λ ∈ L p ( X, µ ) , ∀ h ∈ Λ : | β ( h, − ) | Γ ∈ L p ( Y, ν ) . If an ℓ p -ME-coupling exists, say that Γ and Λ are ℓ p -ME. Clearly any ℓ p -ME-coupling is ℓ q for all q ≤ p .ME-coupling So ℓ − M E is the weakest and ℓ ∞ -ME is the most stringentamong these relations. One can check that ℓ p -ME is an equivalence relation on groups(composition of ℓ p -couplings is ℓ p ), so we obtain a hierarchy of ℓ p -ME categories with ℓ -ME being the weakest (largest classes) and at p = ∞ one arrives at ME+QI. So ℓ p -ME isMeasure Equivalence with some geometric flavor.The setting of ℓ -ME is considered in a joint work with Uri Bader and Roman Sauer [11]to analyze rigidity of the least rigid family of lattices – lattices in SO n, ( R ) ≃ Isom( H n R ), n ≥
3, and fundamental groups of general negatively curved manifolds. Admittedly, for usthe main motivation to considered this restricted type of ME lies in the methods we use. Itshould be noted, however that examples of non-amenable ME groups which are not ℓ -MEseem to be rare (surface groups and free groups seem to be the main culprits). In particular,it follows from Shalom’s computations in [111] that for n ≥ n, ( R ) aremutually ℓ -ME. We shall return to invariants and rigidity in ℓ -ME framework in § § SURVEY OF MEASURED GROUP THEORY 11 Measure Equivalence between groups
This section is concerned with the notion of Measure Equivalence between countablegroups Γ ME ∼ Λ (Definition 2.1). First recall the following deep result (extending previouswork of Dye [26, 27] on some amenable groups, and followed by Connes-Feldman-Weiss [21]concerning all non-singular actions of all amenable groups)
Theorem 3.1 (Ornstein-Weiss [91]) . Any two ergodic probability measure preserving ac-tions of any two infinite countable amenable groups are Orbit Equivalent.
This result implies that all infinite countable amenable groups are ME; moreover forany two infinite amenable groups Γ and Λ there exists an ergodic ME-coupling Ω withindex [Γ : Λ] Ω = 1 (hereafter we shall denote this situation by Γ OE ∼ Λ). Measure Equiva-lence of all amenable groups shows that many QI-invariants are not ME-invariants; theseinclude: growth type, being virtually nilpotent, (virtual) cohomological dimension, finitegenerations/presentation etc.The following are basic constructions and examples of Measure Equivalent groups:(1) If Γ and Λ can be embedded as lattices in the same lcsc group, then Γ ME ∼ Λ.(2) If Γ i ME ∼ Λ i for i = 1 , . . . , n then Γ × · · · × Γ n ME ∼ Λ × · · · × Λ n .(3) If Γ i OE ∼ Λ i for i ∈ I (i.e. the groups admit an ergodic ME-coupling with index one)then ( ∗ i ∈ I Γ i ) OE ∼ ( ∗ i ∈ I Λ i ) .For 2 = n, m < ∞ the free groups F n and F m are commensurable, and therefore are ME(but F ∞ ME ∼ F ). The Measure Equivalence class ME( F ≤ n< ∞ ) is very rich and remainsmysterious (see [49]). For example it includes: surface groups π (Σ g ), g ≥
2, non uniform(infinitely generated) lattices in SL ( F p [[ X ]]), the automorphism group of a regular tree, freeproducts ∗ ni =1 A i of arbitrary infinite amenable groups, more complicated free products suchas F ∗ π (Σ g ) ∗ Q , etc. In the aforementioned paper by Gaboriau he constructs interestinggeometric examples of the form ∗ nc F g , which are fundamental groups of certain ”branchedsurfaces”. Notice that ME( F ≤ n< ∞ ) contains uncountably many groups.The fact that some ME classes are so rich and complicated should emphasize the impres-sive list of ME invariants and rigidity results below.3.1. Measure Equivalence Invariants.
By ME- invariants we mean properties of groupswhich are preserved under Measure Equivalence, and numerical invariants which are pre-served or predictably transformed as a function of the ME index.3.1.1.
Amenability, Kazhdan’s property (T), a-T-menability.
These properties are definedusing the language of unitary representations. Let π : Γ → U ( H ) be a unitary representa-tion of a (topological) group. Given a finite (resp. compact) subset K ⊂ G and ǫ >
0, wesay that a unit vector v ∈ H is ( K, ǫ )- almost invariant if k v − π ( g ) v k < ǫ for all g ∈ K .A unitary Γ-representation π which has ( K, ǫ )-almost invariant vectors for all K ⊂ Γ and ǫ > weakly contain the trivial representation Γ , denoted Γ ≺ π . The trivial The appearance of the sharper condition OE ∼ in (2) is analogous to the one in the QI context: if groupsΓ i and Λ i are bi-Lipschitz then ∗ i ∈ I Γ i QI ∼ ∗ i ∈ I Λ i . representation is (strongly) contained in π , denoted Γ < π , if the subspace of invariants H π (Γ) = { v ∈ H : π (Γ) v = v } is not trivial. We recall: Amenability:
Γ is amenable if the trivial representation is weakly contained in theregular representation ρ : Γ → U ( ℓ (Γ)), ρ ( g ) f ( x ) = f ( g − x ). Property (T):
Γ has property (T) (Kazhdan [68]) if for every unitary Γ-representation π : Γ ≺ π implies Γ < π . Equivalently, if there exist K ⊂ Γ and ǫ > π with ( K, ǫ )-almost invariant vectors, has non-trivialinvariant vectors. For compactly generated groups, another equivalent characteri-zation (Delorme and Guichardet) is that any affine isometric Γ-action on a (real)Hilbert space has a fixed point, i.e., if H (Γ , π ) = { } for any (orthogonal) Γ-representation π . We refer to [14] for details. (HAP): Γ is a-T-menable (or has
Haagerup Approximation Property ) if thefollowing equivalent conditions hold: (i) Γ has a mixing Γ-representation weaklycontaining the trivial one, or (ii) Γ has a proper affine isometric action on a (real)Hilbert space. The class of infinite a-T-menable groups contains amenable groups,free groups but is disjoint from infinite groups with property (T). See [20] as areference.Measure Equivalence allows to relate unitary representations of one group to another.More concretely, let (Ω , m ) be a (Γ , Λ) coupling, and π : Λ → U ( H ) be a unitary Λ-representation. Denote by ˜ H the Hilbert space consisting of equivalence classes (mod nullsets) of all measurable, Λ-equivariant maps Ω → H with square-integrable norm over aΛ-fundamental domain:˜ H = ( f : Ω → H : f ( λx ) = π ( λ ) f ( x ) , Z Ω / Λ k f k < ∞ ) mod null sets . The action of Γ on such functions by translation of the argument, defines a unitary Γ-representation ˜ π : Γ → U ( ˜ H ). This representation is said to be induced from π : Λ → U ( H ) via Ω. (This is analogous to the induction of representations from a lattice to theambient group, possibly followed by a restriction to another lattice).The ME invariance of the properties above (amenability, property (T), Haagerup ap-proximation property) can be deduced from the following observations. Let (Ω , m ) be a(Γ , Λ) ME-coupling, π : Λ → U ( H ) a unitary representation and ˜ π : Γ → U ( ˜ H ) thecorresponding induced representation. Then:(1) If π is the regular Λ-representation on H = ℓ (Λ), then ˜ π on ˜ H can be identifiedwith the Γ-representation on L (Ω , m ) ∼ = n · ℓ (Γ), where n ∈ { , . . . , ∞} .(2) If Λ ≺ π then Γ ≺ ˜ π .(3) If (Ω , m ) is Γ × Λ ergodic and π is weakly mixing (i.e. 1 Λ < π ⊗ π ∗ ) then Γ < ˜ π .(4) If (Ω , m ) is Γ × Λ ergodic and π is mixing (i.e. for all v ∈ H : h π ( h ) v, v i → h → ∞ in Λ) then ˜ π is a mixing Γ-representation.Combining (1) and (2) we obtain that being amenable is an ME-invariant. The deep resultof Ornstein-Weiss [91] and Theorem 2.5 imply that any two infinite countable amenablegroups are ME. This gives: SURVEY OF MEASURED GROUP THEORY 13
Corollary 3.2.
The Measure Equivalence class of Z is the class of all infinite countableamenable groups ME( Z ) = Amen . Bachir Bekka and Alain Valette [13] showed that if Λ does not have property (T) thenit admits a weakly mixing representation π weakly containing the trivial one. By (2) and(3) this implies that property (T) is a ME-invariant (this is the argument in [36, Corollary1.4], see also Zimmer [126, Theorem 9.1.7 (b)]). The ME-invariance of amenability andKazhdan’s property for groups indicates that it should be possible to define these propertiesfor equivalence relations and then relate them to groups. This was indeed done by Zimmer[118, 121] and recently studied in the context of measured groupoids in [8, 9]. We return tothis discussion in § Cost of groups.
The notion of the cost of an action/relation was introduced by Levitt[77] and developed by Damien Gaboriau [44, 45, 47]; the monographs [71] and [70] alsocontain an extensive discussion.Given an (ergodic) probability measure preserving action Γ y ( X, µ ) of a group, definethe cost of the action by cost(Γ y X ) = inf X g ∈ Γ µ ( A g )over all possible collections { A g : g ∈ Γ } of measurable subsets of X with the propertythat for every g ∈ Γ and µ -a.e. x there exist h , . . . , h n ∈ Γ and ǫ , . . . , ǫ n ∈ {− , } so thatdenoting x = x , g i = h ǫ i i · · · h ǫ , x i = g i x = h ǫ i i x i − , one has g = g n = h ǫ n n · · · h ǫ , (cid:26) x i − ∈ A h i if ǫ i = 1 x i − ∈ h i A h − i if ǫ i = − ≤ i ≤ n )In other words, the requirement on the collection { A g } is that (a.e.) arrow x g.x in thegroupoid can be written as a composition of arrows from g : A g → g ( A g ) or their inverses.For essentially free actions, cost(Γ y X ) coincides with cost( R Γ y X ) of the orbit relationas defined in § gx = h ǫ n n · · · h ǫ x rather than g = h ǫ n n · · · h ǫ ). The cost of an action can be turned into a group invariant by setting C ∗ (Γ) = inf X cost(Γ y X ) , C ∗ (Γ) = sup X cost(Γ y X )where the infimum/supremum are taken over all essentially free probability measure pre-serving actions of Γ (we drop ergodicity assumption here; in the definition of C ∗ (Γ) essentialfreeness is also superfluous). Groups Γ for which C ∗ (Γ) = C ∗ (Γ) are said to have fixed price,or prix fixe (abbreviated P.F.). For general groups, Gaboriau defined the cost of a group to be the lower one: C (Γ) = C ∗ (Γ) . To avoid confusion, we shall use here the notation C ∗ (Γ) for general groups, and reserve C (Γ) for P.F. groups only. Question 3.3.
Do all countable groups have property P.F.?
The properties C ∗ = 1, 1 < C ∗ < ∞ , and C ∗ = ∞ are ME-invariants. More precisely: Theorem 3.4. If Γ ME ∼ Λ then, denoting c = [Γ : Λ] Ω for some/any (Γ , Λ) -coupling Ω : C ∗ (Λ) − c · ( C ∗ (Γ) − . We do not know whether the same holds for C ∗ . In [45] this ME-invariance is stated forP.F. groups only. Proof.
Let Ω be a (Γ , Λ)-coupling with Γ y X = Ω / Λ and Λ y Y = Ω / Γ being free, where
X, Y ⊂ Ω are Λ − , Γ − fundamental domains. Given any essentially free p.m.p. actionΛ y Z , consider the (Γ , Λ)-coupling ¯Ω = Ω × Z with the actions g : ( ω, z ) ( gω, z ) , h : ( ω, z ) ( hω, hz ) ( g ∈ Γ , h ∈ Λ) . The actions Γ y ¯ X = ¯Ω / Λ and Λ y ¯ Y = ¯Ω / Γ are Stably Orbit Equivalent with index[Γ : Λ] ¯Ω = [Γ : Λ] Ω = c . Hence (using Theorem 4.7) we have c · (cost( R Γ y ¯ X ) −
1) = cost( R Λ y ¯ Y ) − . While Γ y ¯ X is a skew-product over Γ y X , the action Λ y ¯ Y is the diagonal action on¯ Y = Y × Z . Since ¯ Y = Y × Z has Z as a Λ-equivariant quotient, it follows (by consideringpreimages of any ”graphing system”) thatcost(Λ y ¯ Y ) ≤ cost(Λ y Z ) . Since Λ y Z was arbitrary, we deduce C ∗ (Λ) − ≥ c · ( C ∗ (Γ) − (cid:3) Theorem 3.5 (Gaboriau [44, 45, 47]) . The following classes of groups have P.F.:(1) Any finite group Γ has C ∗ (Γ) = C ∗ (Γ) = 1 − | Γ | .(2) Infinite amenable groups have C ∗ (Γ) = C ∗ (Γ) = 1 .(3) Free group F n , ≤ n ≤ ∞ , have C ∗ ( F n ) = C ∗ ( F n ) = n .(4) Surface groups Γ = π (Σ g ) where Σ g is a closed orientable surface of genus g ≥ have C ∗ (Γ) = C ∗ (Γ) = 2 g − .(5) Amalgamated products Γ = A ∗ C B of finite groups have P.F. with C ∗ (Γ) = C ∗ (Γ) = 1 − ( 1 | A | + 1 | B | − | C | ) . In particular C ∗ (SL ( Z )) = C ∗ (SL ( Z )) = 1 + .(6) Assume Γ , Γ have P.F. then the free product Γ ∗ Γ , and more general amalga-mated free products Λ = Γ ∗ A Γ over an amenable group A , has P.F. with C (Γ ∗ Γ ) = C (Γ ) + C (Γ ) , C (Γ ∗ A Γ ) = C (Γ ) + C (Γ ) − C ( A ) . (7) Products Γ = Γ × Γ of infinite non-torsion groups have C ∗ (Γ) = C ∗ (Γ) = 1 .(8) Finitely generated groups Γ containing an infinite amenable normal subgroup have C ∗ (Γ) = C ∗ (Γ) = 1 .(9) Arithmetic lattices Γ of higher Q -rank (e.g. SL n ≥ ( Z ) ) have C ∗ (Γ) = C ∗ (Γ) = 1 . Note that for an infinite group C ∗ (Γ) = 1 iff Γ has P.F. of cost one. So the content ofcases (2), (7), (8), (9) is C ∗ (Γ) = 1. SURVEY OF MEASURED GROUP THEORY 15
Question 3.6.
Is it true that for all (irreducible) lattices Γ in a (semi-)simple Lie group G of higher rank have P.F. of C ∗ (Γ) = 1 ? (note that they have C ∗ (Γ) = 1 ).Is it true that any group Γ with Kazhdan’s property (T) has P.F. with C ∗ (Γ) = 1 ? Item (9) provides a positive answer to the first question for some non-uniform lattices in higher rank Lie groups, but the proof relies on the internal structure of such lattices(chains of pairwise commuting elements), rather than its relation to the ambient Lie group G (which also has a lot of commuting elements). The motivation for the second question isthat property (T) implies vanishing of the first ℓ -Betti number, β (2)1 (Γ) = 0, while Gaboriaushows that for infinite groups:(3.1) β (2)1 (Γ) = β (2)1 ( R Γ y X ) ≤ cost( R Γ y X ) − β (2)1 (Γ) = 0 (an argument in the spiritof the current discussion is: β (2)1 for ME groups are positively proportional by Gaboriau’sTheorem 3.8, an irreducible lattice in a product is ME to a product of lattices and productsof infinite groups have β (2)1 = 0 by K¨uneth formula. Shalom’s [110] provides a completelygeometric explanation).To give the flavor of the proofs let us indicate the argument for fact (8). Let Γ be a groupgenerated by a finite set { g , . . . , g n } and containing an infinite normal amenable subgroup A and Γ y ( X, µ ) be an essentially free (ergodic) p.m.p. action. Since A is amenable, thereis a Z -action on X with R A y X = R Z y X (mod null sets), and we let φ : X → X denote theaction of the generator of Z . Given ǫ > E ⊂ X with 0 < µ ( E ) < ǫ so that S a ∈ A aE = S φ n E = X mod null sets (if A -action is ergodic any positive measureset works; in general, one uses the ergodic decomposition). For i = 1 , . . . , n let φ i be therestriction of g i to E . Now one easily checks that the normality assumption implies thatΦ = { φ , φ , . . . , φ n } generates R Γ y X , while cost(Φ) = 1 + nǫ .For general (not necessarily P.F.) groups Γ i a version of (6) still holds: C ∗ (Γ ∗ Γ ) = C ∗ (Γ ) + C ∗ (Γ ) , C ∗ (Γ ∗ A Γ ) = C ∗ (Γ ) + C ∗ (Γ ) − C ( A )where A is finite or, more generally, amenable.Very recently Miklos Abert and Benjamin Weiss [2] showed: Theorem 3.7 (Abert-Weiss [2]) . For any discrete countable group Γ , the highest cost C ∗ (Γ) is attained by non-trivial Bernoulli actions Γ y ( X , µ ) Γ and their essentially free quo-tients. Some comments are in order. Kechris [70] introduced the following notion: for probabilitymeasure preserving actions of a fixed group Γ say that Γ y ( X, µ ) weakly contains Γ y Y if given any finite measurable partition Y = F ni =1 Y i , a finite set F ⊂ Γ and an ǫ >
0, thereis a finite measurable partition X = F ni =1 X i so that | µ ( gX i ∩ X j ) − ν ( gY i ∩ Y j ) | < ǫ (1 ≤ i, j ≤ n, g ∈ F ) . The motivation for the terminology is the fact that weak containment of actions implies(but not equivalent to) weak containment of the corresponding unitary representations: L ( Y ) ≺ L ( X ). It is clear that a quotient is (weakly) contained in the larger action. It isalso easy to see that the cost of a quotient action is greater or equal to that of the original(one can lift any graphing from a quotient to the larger action maintaining the cost of thegraphing). Kechris [70] proves that this (anti-)monotonicity still holds in the context ofweak containment:Γ y Y (cid:22) Γ y X = ⇒ cost(Γ y Y ) ≥ cost(Γ y X ) . In fact, it follows from the more general fact that cost is upper semi-continuous in thetopology of actions. Abert and Weiss prove that Bernoulli actions (and their quotients) are weakly contained in any essentially free action of a group. Thus Theorem 3.7 follows fromthe monotonicity of the cost.3.1.3. ℓ -Betti numbers. Cheeger and Gromov [19] associate to every countable group Γ asequence of numbers β (2) i (Γ) ∈ [0 , ∞ ], i ∈ N , which are called ℓ -Betti numbers since theyarise as dimensions (in the sense of Murray von-Neumann) of certain Homology groups(which are Hilbert Γ-modules). For reference we suggest [28], [79]. Here let us just pointout the following facts:(1) If Γ is infinite amenable, then β (2) i (Γ) = 0, i ∈ N ;(2) For free groups β (2)1 ( F n ) = n − β (2) i ( F n ) = 0 for i > β (2)1 (Γ) = 0.(4) K¨uneth formula: β (2) k (Γ × Γ ) = P i + j = k β (2) i (Γ ) · β (2) j (Γ ).(5) Kazhdan’s conjecture, proved by L¨uck, states that for finitely presented residuallyfinite groups ℓ -Betti numbers is the stable limit of Betti numbers of finite indexsubgroups normalized by the index: β (2) i (Γ) = lim β i (Γ n )[Γ:Γ n ] where Γ > Γ > . . . is achain of normal subgroups of finite index.(6) The ℓ Euler characteristic χ (2) (Γ) = P ( − i · β (2) i (Γ) coincides with the usual Eulercharacteristic χ (Γ) = P ( − i · β i (Γ), provided both are defined, as is the case forfundamental group Γ = π ( M ) of a compact aspherical manifold.(7) According to Hopf-Singer conjecture the ℓ -Betti numbers for a fundamental groupΓ = π ( M ) of a compact aspherical manifold M , vanish except, possibly, in the mid-dle dimension n . In particular, ℓ -Betti numbers are integers (Atiyah’s conjecture).The following remarkable result of Damien Gaboriau states that these intricate numericinvariants of groups are preserved, up to rescaling by index, under Measure Equivalence. Theorem 3.8 (Gaboriau [46], [48]) . Let Γ ME ∼ Λ be ME countable groups. Then β (2) i (Λ) = c · β (2) i (Γ) ( i ∈ N ) where c = [Γ : Λ] Ω is a/the index of some/any (Γ , Λ) -coupling. In fact, Gaboriau introduced the notion of ℓ -Betti numbers for II -relations and relatedthem to ℓ -Betti numbers of groups in case of the orbit relation for an essentially free p.m.p.action – see more comments in § ℓ -Betti numbers for fundamental groupsof aspherical manifolds, such as Euler characteristic and sometimes the dimension, pass SURVEY OF MEASURED GROUP THEORY 17 through Measure Equivalence. In particular, if lattices Γ i ( i = 1 ,
2) (uniform or not) inSU n i , ( R ) are ME then n = n ; the same applies to Sp n i , ( R ) and SO n i , ( R ). (The higherrank lattices are covered by stronger rigidity statements – see § D (2) (Γ) = n i ∈ N : 0 < β (2) i (Γ) < ∞ o is a ME-invariant. Conjecture (7) relates this to the dimension of a manifold M in case ofΓ = π ( M ). One shouldn’t expect dim( M ) to be an ME-invariant of π ( M ) as the examplesof tori show; note also that for any manifold M one has π ( M × T n ) ME ∼ π ( M × T k ).However, among negatively curved manifolds Theorem 3.13 below shows that dim( M ) isinvariant under a restricted version of ME.For closed aspherical manifolds M the dimension dim( M ) is a QI invariant of π ( M ).Pansu proved that the whole set D (2) (Γ) is a QI invariant of Γ. However, positive pro-portionality of ℓ -Betti numbers for ME fails under QI; in fact, there are QI groups whoseEuler characteristics have opposite signs. Yet Corollary 3.9.
For ME groups Γ and Λ with well defined Euler characteristic, say funda-mental groups of compact manifolds, one has χ (Λ) = c · χ (Γ) , where c = [Γ : Λ] Ω ∈ (0 , ∞ ) . In particular, the sign (positive, zero, negative) of the Euler characteristic is a ME-invariant.
Cowling-Haagerup Λ -invariant. This mysterious numeric invariant Λ G , taking valuesin [1 , ∞ ], is defined for any lcsc group G in terms of norm bounds on unit approximationin the Fourier algebra A ( G ) (see Cowling and Haagerup [24]). The Λ-invariant coincidesfor a lcsc group and its lattices. Moreover, Cowling and Zimmer [25] proved that Γ OE ∼ Γ implies Λ Γ = Λ Γ . In fact, their proof implies the invariance under Measure Equivalence(see [66]). So Λ Γ is an ME-invariant.Cowling and Haagerup [24] computed the Λ-invariant for simple Lie groups and theirlattices: in particular, proving that Λ G = 1 for G ≃ SO n, ( R ) and SU n, ( R ), Λ G = 2 n − G ≃ Sp n, ( R ), and for the exceptional rank-one group G = F − that Λ G = 21.Hence in the class of simple Lie groups Λ G > G has Kazhdan’s property(T). It is known that Λ G = 1 implies Haagerup property (a.k.a. a-T-menability), but theconverse, conjectured by Cowling, is still open.One may deduce now that if Γ is a lattice in G ≃ Sp n, ( R ) or in F − and Λ is a latticein a simple Lie group H , then Γ ME ∼ Λ iff G ≃ H . Indeed, higher rank H are ruled out byZimmer’s Theorem 3.15; H cannot be in the families SO n, ( R ) and SU n, ( R ) by property(T) or Haagerup property; and within the family of Sp n, ( R ) and F − the Λ-invariantdetects G ( ℓ -Betti numbers can also be used for this purpose).3.1.5. Treeability, anti-treeability, ergodic dimension.
In [4] Scott Adams introduced thenotion of treeable equivalence relations (see § Treeable: if there exists an essentially free p.m.p. Γ-action with a treeable orbitrelation.
Strongly treeable: if every essentially free p.m.p. Γ-action gives a treeable orbitrelation. Anti-treeable: if there are no essentially free p.m.p. Γ-actions with a treeable orbitrelation.Amenable groups and free groups are strongly treeable. It seems to be still unknown whetherthere exist treeable but not strongly treeable groups, in particular it is not clear whethersurface groups (which are treeable) are strongly treeable.The properties of being treeable or anti-treeable are ME-invariants. Moreover, Γ istreeable iff Γ is amenable (i.e. ME ∼ F = Z ), or is ME to either F or F ∞ (this fact usesHjorth’s [59], see [71, Theorems 28.2 and 28.5]). Groups with Kazhdan’s property (T) areanti-treeable [6]. More generally, it follows from the recent work of Alvarez and Gaboriau[7] that a non-amenable group Γ with β (2)1 (Γ) = 0 is anti-treeable (in view of (3.1) thisalso strengthens [45, Corollaire VI.22], where Gaboriau showed that a non-amenable Γ with C (Γ) = 1 is anti-treeable).A treeing of a relation can be seen as a Γ-invariant assignment of pointed trees withΓ as the set of vertices. One may view the relation acting on this measurable family ofpointed trees by moving the marked point. More generally, one might define actions byrelations, or measured groupoids, on fields of simplicial complexes. Gaboriau defines (see[48]) the geometric dimension of a relation R to be the smallest possible dimensionof such a field of contractible simplicial complexes; the ergodic dimension of a groupΓ will be the minimal geometric dimension over orbit relations R Γ y X of all essentiallyfree p.m.p. Γ-actions. In this terminology R is treeable iff it has geometric dimensionone, and a group Γ is treeable if its ergodic dimension is one. there is also a notionof an approximate geometric/ergodic dimension [48] describing the dimensions of asequence of subrelations approximating a given (orbit relation). Theorem 3.10 (Gaboriau [ § . Ergodic dimension and approx-imate ergodic dimension are ME-invariants.
This notion can be used to obtain some information about ME of lattices in the familyof rank one groups SO n, ( R ). If Γ i < SO n i , ( R ), i = 1 , ME ∼ Γ ,then Gaboriau’s result on ℓ -Betti numbers shows that if one of n i is even, then n = n .However, for n i = 2 k i + 1 all β (2) i vanish. In this case Gaboriau shows, using the aboveergodic dimension, that k ≤ k ≤ k or k ≤ k ≤ k .3.1.6. Free products.
It was mentioned above that if Γ i OE ∼ Λ then ∗ i ∈ I Γ i OE ∼ ∗ i ∈ I Λ i (HereΓ OE ∼ Λ means that the two groups admit an ergodic ME-coupling with index one, equiva-lently admit essentially free actions which are orbit equivalent ). To what extent does theconverse hold?In a recent preprint Alvarez and Gaboriau [7] give a very clear answer to this question,for free products of groups which they call measurably freely indecomposable (MFI).They prove that any non-amenable group with β (2)1 = 0 is MFI. Theorem 3.11 (Alvarez-Gaboriau [7]) . Suppose that ∗ ni =1 Γ i ME ∼ ∗ mj =1 Λ j , where { Γ i } ni =1 and { Λ j } mj =1 are two sets of MFI groups with Γ i ME ∼ Γ i ′ for ≤ i = i ′ ≤ n , and Λ j ME ∼ Λ j ′ SURVEY OF MEASURED GROUP THEORY 19 for ≤ j = j ′ ≤ m .Then n = m and, up to a permutation of indices, Γ i ME ∼ Λ i . Another result from [7] concerning decompositions of equivalence relations as free prod-ucts of sub-relations is discussed in § Classes C reg and C . In § second bounded cohomology with unitary coefficients: H b (Γ , π )– a certain vector space associated to a countable group Γ and a unitary representation π : Γ → U ( H π ). (Some background on bounded cohomology can be found in [88, § C reg of groupscharacterized by the property that H b (Γ , ℓ (Γ)) = { } and a larger class C of groups Γ with non-vanishing H b (Γ , π ) for some mixing Γ-representation π . Known examples of groups in C reg ⊂ C include groups admitting ”hyperbolic-like” actionsof the following types: (see [87], [83])(i) non-elementary simplicial action on some simplicial tree, proper on the set of edges;(ii) non-elementary proper isometric action on some proper CAT(-1) space;(iii) non-elementary proper isometric action on some Gromov-hyperbolic graph of boundedvalency.Hence C reg includes free groups, free products of arbitrary countable groups and free prod-ucts amalgamated over a finite group (with the usual exceptions of order two), fundamentalgroups of negatively curved manifolds, Gromov hyperbolic groups, and non-elementarysubgroups of the above families. Examples of groups not in C include amenable groups,products of at least two infinite groups, lattices in higher rank simple Lie groups (over anylocal field), irreducible lattices in products of general compactly generated non-amenablegroups (see [88, § Theorem 3.12 (Monod-Shalom [88]) . (1) Membership in C reg or C is a ME-invariant.(2) For direct products Γ = Γ × · · · × Γ n where Γ i ∈ C reg are torsion free, the numberof factors and their ME types are ME-invariants.(3) For Γ as above, if Λ ME ∼ Γ then Λ cannot be written as product of m > n infinitetorsion free factors. Dimension and simplicial volume ( ℓ -ME). Geometric properties are hard to capturewith the notion of Measure Equivalence. The ℓ -Betti numbers is an exception, but thisinvariant benefits from its Hilbert space nature. In a joint work with Uri Bader and RomanSauer [11] a restricted version of Measure Equivalence, namely ℓ -ME, is considered (see2.3.2 for definition). Being ℓ -ME is an equivalence relation between finitely generatedgroups, in which any two integrable lattices in the same lcsc group are ℓ -ME. All uniformlattices are integrable, and so are all lattices in SO n, ( R ) ≃ Isom( H n R ) (see 3.2.4). Theorem 3.13 (Bader-Furman-Sauer [11]) . Let Γ i = π ( M i ) where M i are closed manifoldswhich admit a Riemannian metric of negative sectional curvature. Assume that Γ and Γ admit an ℓ -ME-coupling Ω . Then dim( M ) = dim( M ) and k M k = [Γ : Γ ] Ω · k M k , where k M i k denotes the simplicial volume of M i . The simplicial volume k M k of a closed manifold M , introduced by Gromov in [56], is thenorm of the image of the fundamental class under the comparison map H n ( M ) → H ℓ n ( M )into the ℓ -homology, which is an ℓ -completion of the usual homology. This is a homotopyinvariant of manifolds. Manifolds carrying a Riemannian metric of negative curvature have k M k > Orbit/Measure Equivalence Rigidity.
By Measure Equivalence results we meanclassification results in the ME category. In the introduction to this section we mentionedthat the ME class ME( Z ) is precisely all infinite amenable groups. The (distinct) classesME( F ≤ n< ∞ ) and ME( F ∞ ) are very rich and resist precise description. However, a lot isknown about more rigid families of groups.3.2.1. Higher rank lattices.
Theorem 3.14 (Zimmer [119]) . Let G and G ′ be center free simple Lie groups with rk R ( G ) ≥ , let Γ < G , Γ ′ < G ′ be lattices and Γ y ( X, µ ) OE ∼ Γ ′ y ( X ′ , µ ′ ) be Orbit Equivalencebetween essentially free probability measure preserving actions. Then G ∼ = G ′ . Moreoverthe induced actions G y ( G × Γ X ) , G ′ y ( G ′ × Γ ′ Y ) are isomorphic up to a choice of theisomorphism G ∼ = G ′ . In other words ergodic (infinite) p.m.p. actions of lattices in distinct higher rank semi-simple Lie groups always have distinct orbit structures , for example2 ≤ n < m = ⇒ SL n ( Z ) y T n OE ∼ SL m ( Z ) y T m . This seminal result (a contemporary of Ornstein-Weiss Theorem 3.1) not only showed thatthe variety of orbit structures of non-amenable groups is very rich, but more importantlyestablished a link between OE in Ergodic Theory and the theory of algebraic groups andtheir lattices; in particular, introducing Margulis’ superrigidity phenomena into ErgodicTheory. This remarkable result can be considered as the birth of the subject discussed inthis survey. Let us record a ME conclusion of the above.
Corollary 3.15 (Zimmer) . Let G , G ′ be connected center free simple Lie groups with rk R ( G ) ≥ , Γ < G and Γ ′ < G ′ lattices. Then Γ ME ∼ Γ ′ iff G ∼ = G ′ . The picture of ME classes of lattices in higher rank simple Lie groups can be sharpenedas follows. There is no need to assume here that the actions are essentially free. This was proved by Stuck andZimmer [115] for all non-atomic ergodic p.m.p. actions of higher rank lattices, generalizing the famous FactorTheorem of Margulis [81], see [82].
SURVEY OF MEASURED GROUP THEORY 21
Theorem 3.16 ([36]) . Let G be a center free simple Lie group with rk R ( G ) ≥ , Γ < G alattice, Λ some group Measure Equivalent to Γ .Then Λ is commensurable up to finite kernels to a lattice in G . Moreover any ergodic (Γ , Λ) -coupling has a quotient which is either an atomic coupling (in which case Γ and Λ are commensurable), or G , or Aut( G ) with the Haar measure. (Recall that Aut( G ) contains Ad( G ) ∼ = G as a finite index subgroup). The main point ofthis result is a construction of a representation ρ : Λ → Aut( G ) for the unknown group Λusing ME to a higher rank lattice Γ. It uses Zimmer’s cocycle superrigidity theorem and aconstruction involving a bi-Γ-equivariant measurable map Ω × Λ ˇΩ → Aut( G ). An updatedversion of this construction is stated in § → Aut( G ) with Φ( γω ) = γ Φ( ω ) , Φ( λω ) = Φ( ω ) ρ ( λ ) − which defines the above quotients (the push-forward measure Φ ∗ m is identified as eitheratomic, or Haar measure on G ∼ = Ad( G ) or on all of Aut( G ), using Ratner’s theorem [103]).This additional information is useful to derive OE rigidity results (see Theorem 4.19).3.2.2. Products of hyperbolic-like groups.
The results above use in an essential way cocyclesuperrigidity theorem of Zimmer, which exploits higher rank phenomena as in Margulis’superrigidity. A particular situation where such phenomena take place are irreducible lat-tices in products of (semi)simple groups, starting from SL ( R ) × SL ( R ); or cocycles over irreducible actions of a product of n ≥ G × · · · × G n y ( X, µ ) means ergodicity of G i y ( X, µ ) for each 1 ≤ i ≤ n . It recentlybecame clear that higher rank phenomena occur also for irreducible lattices in products of n ≥ n ≥ ”higher rank thrust” tothe situation. Of course, one should use factors which are not too small (non-amenable asa minimum). The following break-through results of Nicolas Monod and Yehuda Shalomis an excellent illustration of this fact (see § Theorem 3.17 (Monod-Shalom [88, Theorem 1.16]) . Let
Γ = Γ × · · · × Γ n and Λ =Λ × · · · × Λ m be products of torsion-free countable groups, where Γ i ∈ C reg . Assume that Γ ME ∼ Λ .Then n ≥ m . If n = m then, after a permutation of the indices, Γ i ME ∼ Λ i . In the lattercase ( n = m ) any ergodic ME-coupling of Γ ∼ = Λ has the trivial coupling as a quotient. Theorem 3.18 (Monod-Shalom [88]) . Let
Γ = Γ × · · · × Γ n where n ≥ and Γ i are torsionfree groups in class C , and Γ y ( X, µ ) be an irreducible action (i.e., every Γ i y ( X, µ ) isergodic); let Λ be a torsion free countable group and Λ y ( Y, ν ) be a mildly mixing action.If Γ y X SOE ∼ Λ y Y , then this SOE has index one, Λ ∼ = Γ and the actions are isomorphic. Sometimes this can be relaxed to ergodicity of G ′ i y ( X, µ ) where G ′ i = Q j = i G j . This is relevant in n ≥ Theorem 3.19 (Monod-Shalom [88]) . For i = 1 , let → A i → ¯Γ i → Γ i → be shortexact sequence of groups with A i amenable and Γ i are in C reg and are torsion free. Then ¯Γ ME ∼ ¯Γ implies Γ ME ∼ Γ . The main examples of groups in classes C reg , C belong also to another class of groups D ea which we describe below. Definition 3.20.
A continuous action H y M of a lcsc group (maybe discrete countable)group H on a compact space M is a convergent action if the diagonal H -action on thespace M (3) of disjoint triples is proper. A subgroup H < H is elementary with respect tosuch an action H y M , if H is either pre-compact, or fixes a point, or a pair of points in M ; equivalently, if H has an invariant probability measure on M . Denote by D the class ofgroups H which admit a convergent action where H itself is not elementary . We denoteby D ea ⊂ D the subclass of groups which admit a convergent action where all elementarysubgroups are amenable.Most groups with hyperbolic like behavior, such as: rank one Lie groups and their non-amenable closed subgroups, Gromov hyperbolic groups, groups relatively hyperbolic withrespect to a finite family of amenable groups etc. belong to both C reg and D ea .Monod-Shalom proofs of the results above rely on a cocycle superrigidity theorem 5.5,which involves second bounded cohomology H b of groups, which led them to define theclasses C reg , C . In a joint work with Uri Bader [10] (see also [12]) we develop a differentapproach to higher rank phenomena, in particular showing an analogue 5.6 of Monod-Shalom Theorem 5.5 but refers to class D ea . Plugging this into Monod-Shalom machineryone gets Corollary 3.21.
Theorems 3.17–3.19 remain valid if C reg or C is replaces by D ea . Mapping Class Groups.
The following remarkable result of Yoshikata Kida concernsMapping Class Groups of surfaces. Given a compact orientable surface Σ g,p of genus g with p boundary components the extended mapping class group Γ(Σ g,p ) ⋄ is the groupof isotopy components of diffeomorphisms of Σ g,p (the mapping class group itself is theindex two subgroup of isotopy classes of orientation preserving diffeomorphisms). In thefollowing assume 3 g + p >
0, i.e., rule out the torus Σ , , once punctured torus Σ , , andspheres Σ ,p with p ≤ Theorem 3.22 (Kida [73]) . Let Γ be a finite index subgroup in Γ(Σ g,p ) ⋄ with g + p − > ,or in a finite product of such Mapping Class groups Q ni =1 Γ(Σ g,p ) ⋄ .Then any group Λ ME ∼ Γ is commensurable up to finite kernels to Γ , and ergodic ME-coupling has a discrete (Γ , Λ) -coupling as a quotient. This work (spanning [72–75]) is a real tour de force. Mapping Class Groups Γ(Σ) areoften compared to a lattice in a semi-simple Lie group G : the Teichm¨uller space T (Σ) isanalogous to the symmetric space G/K , Thurston boundary S (Σ) analogous to Furstenbergboundary B ( G ) = G/P , and the curve complex C (Σ) to the spherical Tits building of G . This class was introduced by Furstenberg in [42], in the current literature these groups are also knownas convergent groups . Furstenberg used the term D -groups after Dynkin who studied the action of thefree group on the space of ends of the tree. SURVEY OF MEASURED GROUP THEORY 23
The MCG has been extensively studied as a geometric object, while Kida’s work provides anew Ergodic theoretic perspective. For example, Kida proves that Thurston boundary withthe Lebesgue measure class is Γ- boundary in the sense of Burger-Monod for the MappingClass Group, i.e., the action of the latter is amenable and doubly ergodic with unitarycoefficients. Properties of the MCG action on Thurston boundary allow Kida to characterizecertain subrelations/subgroupoids arising in self Measure Equivalence of a MCG; leadingto the proof of a cocycle (strong) rigidity theorem 5.7, which can be viewed as a groupoidversion of Ivanov’s rigidity theorem. This strong rigidity theorem can be used with § Hyperbolic lattices and ℓ -ME. Measure Equivalence is motivated by the theory oflattices, with ME-couplings generalizing the situation of groups imbedded as lattices inin the same ambient lcsc group. Thus, in the context of semi-simple groups, one wonderswhether ME rigidity results would parallel Mostow rigidity; and in particular would apply to(lattices in) all simple groups with the usual exception of SL ( R ) ≃ SO , ( R ) ≃ SU , ( R ).Higher rank situation (at least that of simple groups) is understood ( § n, ( R ), SU m, ( R ), Sp k, ( R ), and F − ) known ME-invariants discussed above (property (T), ℓ -Betti numbers, Λ-invariant, ergodic dimension)allow to distinguish lattices among most of rank one groups. This refers to statements ofthe form: if Γ i < G i are lattices then Γ ME ∼ Γ iff G ≃ G . However ME classification suchas in Theorems 3.16, 3.17, 3.22 are not known (yet?) for rank one cases. The ingredientwhich is missing in the existing approach is an appropriate cocycle superrigidity theorem .In a joint work with Uri Bader and Roman Sauer a cocycle strong rigidity theoremis proved for ME-cocycles for lattices in SO n, ( R ) ≃ Isom( H n R ), n ≥
3, under a certain ℓ -assumption (see § Theorem 3.23 (Bader-Furman-Sauer [11]) . Let Γ is a lattice in G = Isom( H n ) , n ≥ , and Λ is some finitely generated group ℓ -ME to Γ then Λ is a lattice in G modulo a finite normalsubgroup. Moreover any ergodic (Γ , Λ) -coupling has a quotient, which is ether discrete, or G = Aut( G ) , or G with the Haar measure. Recently Sorin Popa has introduced a new set of ideas for studying Orbit Equivalence.These results, rather than relying on rigidity of the acting groups alone, exploit rigidityaspects of groups actions of certain type. We shall discuss them in §§ How many Orbit Structures does a given group have?
Theorem 3.1 of Ornsteinand Weiss [91] implies that for an infinite amenable countable group Γ all ergodic probabilitymeasure preserving actions Γ y ( X, µ ) define the same orbit structure, namely R amen .What happens for non-amenable groups Γ? The very recent answer to this question is For Sp n, ( R ) and F − a cocycle superrigidity theorem was proved by Corlette and Zimmer [23], butthis result requires boundness assumptions which preclude it from being used for ME-cocycles. Theorem 3.24 (Epstein [29]) . Any non-amenable countable group Γ has a continuum ofessentially free ergodic probability measure preserving actions Γ y ( X, µ ) , no two of whichare weakly Orbit Equivalent. Let us briefly discuss the problem and its solution. Since Card(Aut(
X, µ ) Γ ) = ℵ = 2 ℵ there are at most continuum many (in fact, precisely that many) actions for a fixed countablegroup Γ. Hence one might expect at most ℵ -many non-OE actions for a given non-amenableΓ. Most invariants of equivalence relations depend on the acting group rather than theaction, and thus could not help to distinguish actions of the fixed group Γ. Most, butnot all, there are a few exceptions. For non-amenable groups which do not have property(T) two non- SOE actions can easily be constructed: (1) a strongly ergodic action (usingSchmidt’s [109]) and (2) an ergodic action which is not strongly ergodic (using Connes-Weiss[22]). Taking a product with an essentially free weakly mixing strongly ergodic Γ-actions(e.g. the Bernoulli action ( X , µ ) Γ ) one makes the above two actions essentially free anddistinct.OE rigidity results showed that some specific classes of non-amenable groups have un-countably many mutually non-OE actions: this includes higher rank lattices [55], productsof hyperbolic-like groups [88, Theorem 1.7], further classes of groups [93, 96]). But thegeneral question remained open.In [58] Greg Hjorth showed that for Γ with property (T) the set of isomorphism classesof orbit structures for essentially free Γ-actions has cardinality ℵ . This beautiful argumentcombines local rigidity (see § separability argument, analogues of whichappeared in Sorin Popa’s work in von Neumann algebras, to show that the natural mapfrom isomorphism classes of Γ-actions to isomorphism classes of Γ-orbit structures is atmost countable-to-one. Since the former set has cardinality of a continuum ℵ , so is thelatter.The next big advance was made by Damien Gaboriau and Sorin Popa in [50], wherethey produced a continuum of essentially free actions of the free group F where the mapfrom actions to orbit structures is countable-to-one. Here the rigidity component usedPopa’s notion of w -rigid actions such as SL ( Z )-action on the torus T (the rigidity comingfrom the relative property (T) for the semi-direct product SL ( Z ) ⋉ Z viewing Z as thePontryagin dual of T ). In [61] Adrian Ioana provided an explicit list of a continuum ofmutually non-SOE actions of F .In [63] Adrian Ioana leaped further by proving that any Γ containing F has uncountablymany mutually non-SOE essentially free actions, by co-inducing F -actions on X to Γ y X Γ / F and pushing the solution of F -problem to the analysis of the co-induced actions.However, this beautiful result did not settle the problem completely, because there existnon-amenable groups not containing a copy of F . The general problem remained open onlyfor a short while. Damien Gaboriau and Russel Lyons [51] proved that any non-amenable Γcontains a F on the level of equivalence relations. Using this fact, Inessa Epstein [29] wasable to show that such containment suffices to carry out an analogue of the co-inductionargument [63].Furthermore, in [64] Ioana, Kechris, Tsankov jointly with Epstein show that for anynon-amenable Γ the space of all ergodic free p.m.p. actions taken up to OE not only hascardinality of the continuum, but is also impossible to classify in a very strong sense. SURVEY OF MEASURED GROUP THEORY 25
Mostow-type rigidity: ME and QI approaches.
A common theme in the theoryof lattices is the study of the relationship between a lattice Γ and the ambient group G .One of the natural questions here is to what extent does the abstract group Γ know itsambient group G – its envelop in the terminology of Furstenberg’s [42]. The latter paperis one of the earliest results along these lines showing that a lattice in rank one simple Liegroup cannot be embedded as a lattice in higher rank semi-simple Lie group. Mostow’sstrong rigidity implies that an (irreducible) lattice Γ in a semi-simple Lie group G cannotbe embedded in any other (not locally isomorphic) semi-simple Lie group H as a lattice;furthermore, there is only one embedding Γ → G as a lattice (up to automorphisms of G ),unless G ≃ SL ( R ). The following is therefore a natural Problem 3.25.
Given a countable group Γ determine all its envelops: lcsc groups H andimbeddings Γ → H , where ρ (Γ) is a lattice in H . A variant of this question concerns uniform envelops , i.e., lcsc groups H containing a copy of Γ as a uniform lattice. Note that Γ is a lattice in itself and in discrete groups containing it as a finite index sub-group. Given any envelop H for Γ, one can construct several related envelopes using compactgroups: H × K , or H ⋉ K , or finite/compact central extensions of H etc. A non-obviousexample for a (uniform) envelop for a finitely generated group is H = Aut(Cayley(Γ , S ))where Cayley(Γ , S ) is the Cayley graph of Γ with respect to some finite generating set S .For example, Aut(Tree k ) is an envelope for F k .Envelopes for lattices in simple Lie group are classified in [38] (general envelops for higherrank lattices, and uniform ones for rank one cases), and envelopes for Mapping Class Groupsby Kida in [73] (Farb and Weinberger [31] study a more geometric question of isometries ofTeichm¨uller space). The answer in these situations is that there are no non-obvious envelops.In particular Aut(Cayley(Γ , S )) is discrete and contains Γ as a finite index subgroup (theindex is bounded independently of S ). Let us explain two approaches to this problem: aME approach for classification of general envelops, and a QI approach to uniform envelops.Recall that a countable group Γ is said to be ICC (Infinite Conjugacy Classes) if { e } isthe only finite conjugacy class of Γ. Consider the following notion. Definition 3.26.
A subgroup Γ in a lcsc group G is strongly ICC if the only probabilitymeasure µ on G which is invariant under the conjugation action γ : g γgγ − ( γ ∈ Γ) isthe Dirac measure at the identity: µ = δ e .A countable ICC group is strongly ICC in itself. More generally, for countable groupsΓ < G strong ICC is equivalent to the property that every finite index subgroup of Γ hastrivial centralizer in G . For a semi-simple real Lie group G with finite center and no non-trivial compact factors, any Zariski dense subgroup Γ < G is strongly ICC in G . Thisremains true in the context of semi-simple algebraic groups. Proposition 3.27.
Let Γ be a countable group, G , H lcsc groups and i : Γ → G , j : Γ → H lattice imbeddings. Assume that i (Γ) is strongly ICC in G and that there exists a measurablemap Φ : H → G satisfying a.e. Φ( j ( γ ) hj ( γ ) − ) = i ( γ )Φ( h ) i ( γ ) − ( γ , γ ∈ Γ) . Borel’s density theorem asserts that all lattices are Zariski dense.
Then Φ coincides a.e. with a continuous homomorphism π : H → G with compact kerneland closed image and j = π ◦ i . In case of higher rank lattice Γ one uses Theorem 5.4, and for Mapping Class GroupsKida relies on Theorem 5.7.
Proof.
Consider a measurable map F : H × H → G given by F ( x, y ) = Φ( x )Φ( x − y )Φ( y ) − Since F ( xj ( γ ) , y ) = F ( x, y ) = F ( x, yj ( γ )) the map F descends to f : X × X → G , where X = H/j (Γ), and satisfies f ( j ( γ ) .x, j ( γ ) .x ′ ) = i ( γ ) f ( x, x ′ ) i ( γ ) − . Thus the pushforwardprobability measure µ = f ∗ ( m H/j (Γ) × m H/j (Γ) ) on G is invariant under conjugation by i (Γ),hence is the Dirac measure at e ∈ G by the assumption. We conclude that Φ : H → G isan a.e. homomorphism, i.e., a.e. on H × H Φ( x − y ) = Φ( x ) − Φ( y ) . A measurable homomorphism between lcsc groups is known (cf. Zimmer [126, AppendixB]) to coincide a.e. with a continuous homomorphism π : H → G . It follows that j = π ◦ i . Compactness of ker( π ) follows from the fact that the pre-image π − ( F ) of a Borelfundamental domain F ⊂ G for j (Γ) has positive but finite Haar measure in H . (cid:3) The problem of uniform lattices can be tackled using Quasi-Isometry. If ρ (Γ) < H is cocompact, then Γ and H are quasi-isometric (assuming Γ is finitely generated, H iscompactly generated and one can equip both with left invariant ”word” (quasi)-metrics).The isometric actions of H on itself gives a homomorphism π : H −→ Isom( H ) → QI( H ) −→ QI(Γ) . Hence a representation H → QI(Γ) comes easily. It remains to analyze the image, thekernel, and check continuity (see [38]).
SURVEY OF MEASURED GROUP THEORY 27 Ergodic equivalence relations
Basic definitions.
We start with the notion of countable equivalence relations in the Borel setting. It consists of a standard Borel space ( X, X ) (cf. [32] for definitions)and a Borel subset R ⊂ X × X which is an equivalence relation, whose equivalence classes R [ x ] = { y ∈ X : ( x, y ) ∈ R } are all countable.To construct such relations choose a countable collection Φ = { φ i } i ∈ I of Borel bijections φ i : A i → B i between Borel subsets A i , B i ∈ X , i ∈ I ; and let R Φ be the smallest equivalencerelation including the graphs of all φ i , i ∈ I . More precisely, ( x, y ) ∈ R Φ iff there exists afinite sequence i , . . . , i k ∈ I and ǫ , . . . , ǫ k ∈ {− , } so that y = φ ǫ k i k ◦ · · · ◦ φ ǫ i ◦ φ ǫ i ( x ) . We shall say that the family Φ generates the relation R Φ . The particular case of a collectionΦ = { φ i } of Borel isomorphisms of the whole space X generates a countable group Γ = h Φ i and R Φ = R Γ y X = { ( x, y ) : Γ x = Γ y } = { ( x, γ.x ) : x ∈ X, γ ∈ Γ } . Feldman and Moore [32] proved that any countable Borel equivalence relation admits agenerating set whose elements are defined on all of X ; in other words, any equivalencerelation appears as the orbit relation R Γ y X of a Borel action Γ y X of some countablegroup Γ (see 4.3.1).Given a countable Borel equivalence relation R the full group [ R ] is defined by[ R ] = { φ ∈ Aut( X, X ) : ∀ x ∈ X : ( x, φ ( x )) ∈ R } . The full pseudo-group [[ R ]] consists of partially defined Borel isomorphisms ψ : Dom( ψ ) → Im( ψ ) with ( x, ψ ( x )) ∈ R for all x ∈ Dom( ψ ).If R is the orbit relation R Γ y X of a group action Γ y ( X, X ), then any φ ∈ [ R ] has thefollowing ”piece-wise Γ-structure”: there exist countable partitions F A i = X = F B i intoBorel sets and elements γ i ∈ Γ with γ i ( A i ) = B i so that φ ( x ) = γ i x for x ∈ A i . Elements ψ of the full pseudo-group [[ R Γ ]] have a similar ”piece-wise Γ-structure” with F A i = Dom( ψ )and F B i = Im( ψ ).Let R be a countable Borel equivalence relation on a standard Borel space ( X, X ). Ameasure µ on ( X, X ) is R - invariant (resp. R - quasi-invariant ) if for all φ ∈ [ R ], φ ∗ µ = µ (resp. φ ∗ µ ∼ µ ). Note that if Φ = { φ i : A i → B i } is a generating set for R then µ is R -invariant iff µ is invariant under each φ i , i.e. µ ( φ − i ( E ) ∩ A i ) = µ ( E ∩ B i ) forall E ∈ X . Similarly, quasi-invariance of a measure can be tested on a generating set.The R - saturation of E ∈ X is R [ E ] = { x ∈ X : ∃ y ∈ E, ( x, y ) ∈ R } . A R (quasi-)invariant measure µ is ergodic if R [ E ] is either µ -null or µ -conull for any E ∈ X . Inthis section we shall focus on countable Borel equivalence relations R on ( X, X ) equippedwith an ergodic, invariant, non-atomic, probability measure µ on ( X, X ). Such a quadruple( X, X , µ, R ) is called type II -relation. These are precisely the orbit relations of ergodicmeasure preserving actions of countable groups on non-atomic standard probability measurespaces (the non-trivial implication follows from the above mentioned theorem of Feldmanand Moore). Given a countable Borel relation R on ( X, X ) and an R -quasi-invariant probability mea-sure µ , define infinite measures ˜ µ L , ˜ µ R on R by˜ µ L ( E ) = Z X { y : ( x, y ) ∈ E ∩ R } dµ ( x ) , ˜ µ R ( E ) = Z X { x : ( x, y ) ∈ E ∩ R } dµ ( y ) . These measures are equivalent, and coincide ˜ µ L = ˜ µ R = ˜ µ if µ is R -invariant, which is ourmain focus. Hereafter, saying that some property holds a.e. on R would refer to ˜ µ -a.e.(this makes sense even if µ is only R -quasi-invariant). Remark 4.1.
In some situations a Borel Equivalence relation R on ( X, X ) has only one(non-atomic) invariant probability measure. For example, this is the case for the orbitrelation of the standard action of a finite index subgroup Γ < SL n ( Z ) on the torus T n = R n / Z n , or for a lattice Γ in a simple center free Lie group G acting on H/ Λ, where H isa simple Lie group, Λ < H is a lattice, and Γ acts by left translations via an embedding j : G → H with j ( G ) having trivial centralizer in H . In such situations one may gainunderstanding of the countable Borel equivalence relation R via the study of the II -relationcorresponding to the unique R -invariant probability measure.As always in the measure-theoretic setting null sets should be considered negligible. Soan isomorphism T between (complete) measure spaces ( X i , X i , µ i ), i = 1 ,
2, is a Borelisomorphism between µ i -conull sets T : X ′ → X ′ with T ∗ ( µ ) = µ . In the context of II -relations, we declare two relations ( X i , X i , µ i , R i ), i = 1 , isomorphic , if the existsa measure space isomorphism T : ( X , µ ) ∼ = ( X , µ ) so that T × T : ( R , ˜ µ ) → ( R , ˜ µ )is an isomorphism. In other words, after a restriction to conull sets, T satisfies( x, y ) ∈ R ⇐⇒ ( T ( x ) , T ( y )) ∈ R . Let us also adapt the notions of the full group and the full pseudo-group to the measure-theoretic setting, by passing to a quotient Aut( X, X ) → Aut( X, X , µ ) where two Borelisomorphism φ and φ ′ which agree µ -a.e. are identified. This allows us to focus on theessential measure-theoretic issues. The following easy, but useful Lemma illustrates theadvantage of this framework. Lemma 4.2.
Let ( X, X , µ, R ) be a II -relation. For A, B ∈ X there exists φ ∈ [ R ] with φ ( A ) = B iff µ ( A ) = µ ( B ) . Note that since the full group [ R ] consists of equivalence classes of Borel maps, theequality φ ( A ) = B means that that there exists a representative f ∈ Aut( X, X ) in the Borelfull group of the Borel relation R , and sets A ′ , B ′ ∈ X with µ ( A △ A ′ ) = µ ( B △ B ′ ) = 0 and f ( A ′ ) = B ′ , or equivalently µ ( f ( A ) △ B ) = 0. Proof.
The ”only if” direction is immediate from µ being R -invariant. So we should showthat given A, B ∈ X of the same µ -measure there exists a φ ∈ [ R ] taking A to B , modulonull sets. The case of µ -null sets being trivial, assume s = µ ( A ) = µ ( B ) >
0. By Feldman-Moore’s theorem there is an ergodic m.p. action Γ y ( X, X , µ ) of a countable group Γ Or just Zariski dense subgroup, see [15].
SURVEY OF MEASURED GROUP THEORY 29 with R = R Γ y X . Enumerate its elements Γ = { g , g , . . . } with g = e . By ergodicity,there exists g ∈ Γ with µ ( gA ∩ B ) >
0. Let t = sup { µ ( gA ∩ B ) : g ∈ Γ } > n = min { n : µ ( g n A ∩ B ) > t } . So g n maps a subset A ⊂ A of size µ ( A ) > t to asubset B = g n ( A ) ⊂ B of the same size. Denoting by A ′ = A − A , B ′ = B − B theremainders, we have µ ( A ′ ) = µ ( B ′ ). If these are null sets stop. Otherwise proceed induc-tively with t i +1 = sup { µ ( gA ′ i ∩ B i ) : g ∈ Γ } and n i +1 = min { n : µ ( g n A ′ i ∩ B i ) > t i +1 } ,and A ′ i +1 ⊂ A ′ i , B ′ i +1 ⊂ B ′ i denoting the yet unmatched remainders. This procedure maystop at some finite time when A ′ i and B ′ i are null sets, or would continue for all i ∈ N .The key point is that in the latter case t i → A = S A i is a subset of A with µ ( A − ˜ A ) = 0 (and the same goes for B ). We obtain ψ ∈ [[ R ]] with ψ : ˜ A → ˜ B definedby ψ ( x ) = g n i .x for x ∈ A i . Applying the same argument to the compliments, we get ψ c ∈ [[ R ]] mapping A c to B c , and patching ψ with ψ c gives the desired element φ ∈ [ R ] ofthe full group in the measure-theoretic sense! (cid:3) Restriction and weak isomorphisms.
Equivalence relations admit a natural operationof restriction , sometimes called induction , to a subset: given a relation R on X and ameasurable subset A ⊂ X the restriction R A to A is R A = R ∩ ( A × A ) . In the presence of, say R -invariant, measure µ on ( X, X ) the restriction to a subset A ⊂ X with µ ( A ) > µ | A , defined by µ | A ( E ) = µ ( A ∩ E ). If µ isa probability measure, we shall denote by µ A the normalized restriction µ A = µ ( A ) − · µ | A .It is easy to see that ergodicity is preserved, so a restriction of a II -relation ( X, µ, R ) to apositive measure subset A ⊂ X is a II -relation ( A, µ A , R A ). Remark 4.3.
Note that it follows from Lemma 4.2 that the isomorphism class of R A depends only on R and on the size µ ( A ), so R A may be denoted R t where t = µ ( A ) is0 < t ≤
1. One may also define R t for t >
1. For an integer k > R k denote the productof R with the full relation on the finite set { , . . . , k } , namely the relation on X × { , . . . , k } with (( x, i ) , ( y, j )) ∈ R k iff ( x, y ) ∈ R . So ( R k ) /k ∼ = R ∼ = R . The definition of R t cannow be extended to all 0 < t < ∞ using an easily verified formula ( R t ) s ∼ = R ts . Thisconstruction is closely related to the notion of an amplification in von-Neumann algebras:the Murray von Neumann group-measure space construction M R satisfies M R t = ( M R ) t .The operation of restriction/induction allows one to relax the notion of isomorphism ofII -relations as follows: Definition 4.4.
Two II -relations R and R are weakly isomorphic if R ∼ = R t for some t ∈ R × + . Equivalently, if there exist positive measurable subsets A i ⊂ X i with µ ( A ) = t · µ ( A ) and an isomorphism between the restrictions of R i to A i .Observe that two ergodic probability measure-preserving actions Γ i y ( X i , X i , µ i ) ofcountable groups are orbit equivalent iff the corresponding orbit relations R Γ i y X i are iso-morphic.4.2. Invariants of equivalence relations.
Let us now discuss in some detail severalqualitative and numerical properties of II equivalence relations which are preserved underisomorphisms and often preserved or rescaled by the index under weak isomorphisms. We refer to such properties as invariants of equivalence relations. Many of these propertiesare motivated by properties of groups, and often an orbit relation R Γ y X of an essentiallyfree action of countable group would be a reflection of the corresponding property of Γ.4.2.1. Amenability, strong ergodicity, property (T).
Amenability of an equivalence relationcan be defined in a number of ways. In [118] Zimmer introduced the notion of amenability for a group action on a space with quasi-invariant measure. This notion plays a central rolein the theory. This definition is parallel to the fixed point characterization of amenabilityfor groups. For equivalence relation R on ( X, X ) with a quasi-invariant measure µ it readsas follows.Let E be a separable Banach space, and c : R → Isom( E ) be a measurable 1- cocycle ,i.e., a measurable (with respect to the weak topology on E ) map, satisfying ˜ µ -a.e. c ( x, z ) = c ( x, y ) ◦ c ( y, z ) . Let X ∋ x Q x ⊂ E ∗ be a measurable family of non-empty convex compact subsets ofthe dual space E ∗ taken with the ∗ -topology, so that c ( x, y ) ∗ ( Q x ) = Q y . The relation R is amenable if any such family contains a measurable invariant section, i.e., a measurableassignment X ∋ x p ( x ) ∈ Q x , so that a.e. c ( x, y ) ∗ p ( x ) = p ( y ) . The (original) definition of amenability for group actions concerned general cocycles c : G × X → Isom( E ) rather than the ones depending only on the orbit relation R Γ y X . Thelanguage of measured groupoids provides a common framework for both settings (see [9]).Any non-singular action of an amenable group is amenable, because any cocycle c :Γ × X → Isom( E ) can be used to define an affine Γ-action on the closed convex subset of L ∞ ( X, E ∗ ) = L ( X, E ) ∗ consisting of all measurable sections x → p ( x ) ∈ Q x ; the fixedpoint property of Γ provides the desired c ∗ -invariant section. The converse is not true: any(countable, or lcsc group) admits essentially free amenable action with a quasi-invariantmeasure – this is the main use of the notion of amenable actions. However, for essentially free, probability measure preserving actions, amenability of the II -relation R Γ y X implies(hence is equivalent to) amenability of Γ. Indeed, given an affine Γ action α on a convexcompact Q ⊂ E ∗ , one can take Q x = Q for all x ∈ X and set c ( gx, x ) = α ( g ); amenability of R Γ y X provides an invariant section p : X → Q whose barycenter q = R X p ( x ) dµ ( x ) wouldbe an α (Γ)-fixed point in Q .Connes, Feldman and Weiss [21] proved that amenable relations are hyperfinite in thesense that they can be viewed as an increasing union of finite subrelations; they also showedthat such a relation can be generated by an action of Z (see also [69] by Kaimanovich fora nice exposition and several other nice characterizations of amenability). It follows thatthere is only one amenable II -relation, which we denote hereafter by R amen . In [121] Zimmer introduced the notion of property (T) for group actions on measure spaceswith quasi-invariant measure. The equivalence relation version can be stated as follows. Let H be a separable Hilbert space and let c : R → U ( H ) be a measurable 1- cocycle , i.e., c satisfies c ( x, z ) = c ( x, y ) ◦ c ( y, z ) SURVEY OF MEASURED GROUP THEORY 31
Then R has property (T) if any such cocycle for which there exists a sequence v n : X → S ( H ) of measurable maps into the unit sphere S ( H ) with k v n ( y ) − c ( x, y ) v n k → µ ]-a.e.admits a measurable map u : X → S ( H ) with u ( y ) = c ( x, y ) u ( x ) for ˜ µ -a.e. ( x, y ) ∈ R .For an essentially free probability measure preserving action Γ y ( X, µ ) the orbit relation R Γ y X has property (T) if and only if the group Γ has Kazhdan’s property (T) (in [121] weakmixing of the action was assumed for the ”only if” implication, but this can be removedas in § R be a II -equivalence relation on ( X, µ ). A sequence { A n } of measurable subsetsof X is asymptotically R -invariant , if µ ( φ ( A n ) △ A n ) → φ ∈ [ R ]. Thisis satisfied trivially if µ ( A n ) · (1 − µ ( A n )) →
0. Relation R is strongly ergodic if anyasymptotically R -invariant sequence of sets is trivial in the above sense. (Note that thecondition of asymptotic invariance may be checked on elements φ i of any generating systemΦ of R ).The amenable relation R amen is not strongly ergodic. If an action Γ y ( X, µ ) has a spectral gap (i.e., does not have almost invariant vectors) in the Koopman representationon L ( X, µ ) ⊖ C then R Γ y X is strongly ergodic. Using the fact that the Koopman rep-resentation of a Bernoulli action Γ y ( X , µ ) Γ is a multiple of the regular representation ∞ · ℓ (Γ), Schmidt [108] characterized non-amenable groups by the property that they ad-mit p.m.p. actions with strongly ergodic orbit relation. If R is not strongly ergodic then ithas an amenable relation as a non-singular quotient (Jones and Schmidt [67]). Connes andWeiss [22] showed that all p.m.p. actions of a group Γ have strongly ergodic orbit relationsif and only if Γ has Kazhdan’s property (T). In this short elegant paper they introducedthe idea of Gaussian actions as a way of constructing a p.m.p. action from a given unitary representation .4.2.2.
Fundamental group - index values of self similarity.
The term fundamental group of a II -relation R refers to a subgroup of R × + defined by F ( R ) = (cid:8) t ∈ R × + : R ∼ = R t (cid:9) . Equivalently, for R on ( X, µ ), the fundamental group F ( R ) consists of all ratios µ ( A ) /µ ( B )where A, B ⊂ X are positive measure subsets with R A ∼ = R B (here one can take one ofthe sets to be X without loss of generality). The notion is borrowed from similarly definedconcept of the fundamental group of a von Neumann algebra, introduced by Murray andvon Neumann [90]: F ( M ) = (cid:8) t ∈ R × + : M t ∼ = M (cid:9) . However, the connection is not direct:even for group space construction M = Γ ⋉ L ∞ ( X ) isomorphisms M ∼ = M t (or evenautomorphisms of M ) need not respect the Cartan subalgebra L ∞ ( X ) in general.Since the restriction of the amenable relation R amen to any positive measure subset A ⊂ X is amenable, it follows F ( R amen ) = R × + . The same obviously applies to the product of any relation with an amenable one.
On another extreme are orbit relations R Γ y X of essentially free ergodic action of ICCgroups Γ with property (T): for such relations the fundamental group F ( R Γ y X ) is at mostcountable (Gefter and Golodets [55, Corollary 1.8]).Many relations have trivial fundamental group. This includes all II relations with anon-trivial numeric invariant which scales under restriction:(1) Relations with 1 < cost( R ) < ∞ ; in particular, orbit relation R Γ y X for essentiallyfree actions of F n , 1 < n < ∞ , or surface groups.(2) Relations with some non-trivial ℓ -Betti number 0 < β (2) i ( R ) < ∞ for some i ∈ N ;in particular, orbit relation R Γ y X for essentially free actions of a group Γ with0 < β (2) i (Γ) < ∞ for some i ∈ N , such as lattices in SO n, ( R ), SU m, ( R ), Sp k, ( R ).Triviality of the fundamental group often appears as a by-product of rigidity of groups andgroup actions. For examples F ( R Γ y X ) = { } in the following situations:(1) Any (essentially free) action of a lattice Γ in a simple Lie group of higher rank ([55]);(2) Any essentially free action of (finite index subgroups of products of) Mapping ClassGroups ([73]);(3) Actions of Γ = Γ × · · · × Γ n , n ≥
2, of hyperbolic-like groups Γ i where each of themacts ergodically ([88]);(4) G dsc -cocycle superrigid actions Γ y X such as Bernoulli actions of groups withproperty (T) ([94, 95, 100]).What are other possibilities for the fundamental group beyond the two extreme cases F ( R ) = R × + and F ( R ) = { } ? The most comprehensive answer (to date) to this questionis contained in the following very recent result of S.Popa and S.Vaes (see [101] for furtherreferences): Theorem 4.5 (Popa-Vaes, [101, Thm 1.1]) . There exists a family S of additive subgroupsof R which contains all countable groups, and (uncountable) groups of arbitrary Hausdorffdimension in (0 , , so that for any F ∈ S and any totally disconnected locally compactunimodular group G there exist uncountably many mutually non-SOE essentially free p.m.p.actions of F ∞ whose orbit relations R = R F ∞ y X have F ( R ) ∼ = exp( F ) and Out( R ) ∼ = G .Moreover, in these examples the Murray von Neumann group space factor M = Γ ⋊ L ∞ ( X ) has F ( M ) ∼ = F ( R ) ∼ = exp( F ) and Out( M ) ∼ = Out( R ) ⋉ H ( R , T ) , where H ( R , T ) is the first cohomology with coefficients in the -torus. Treeability.
An equivalence relation R is said treeable (Adams [4]) if it admits agenerating set Φ = { φ i } so that the corresponding (non-oriented) graph on a.e. R -classis a tree. Basic examples of treeable relations include: R amen viewing the amenable II -relation as the orbit relation of some/any action of Z = F , and more general R F n y X where F n y X is an essentially free action of the free group F n , 1 ≤ n ≤ ∞ . Any restriction of atreeable relation is treeable, and R is treeable iff R t is.If R → R is a (weak) injective relation morphism and R is treeable, then so is R –the idea is to lift a treeing graphing from R to R piece by piece. This way, one showsthat if a group Λ admits an essentially free action Λ y Z with treeable R Λ y Z , and Γand Λ admit (S)OE essentially free actions Γ y X and Λ y Y then the Γ-action on X × Z , g : ( x, z ) ( gx, α ( g, x ) z ) via the (S)OE cocycle α : Γ × X → Λ, has a treeable
SURVEY OF MEASURED GROUP THEORY 33 orbit structure R Γ y X × Z . Since surface groups Γ = π (Σ g ), g ≥
2, and F are lattices inPSL ( R ), hence ME, the former groups have free actions with treeable orbit relations. Areall orbit relations of free actions of a surface group treeable?4.2.4. Cost.
The notion of cost for II -relations corresponds to the notion of rank for dis-crete countable groups. The notion of cost was introduced by G. Levitt [77] and extensivelystudied by D. Gaboriau [44, 45, 51]. Definition 4.6.
Given a generating system Φ = { φ i : A i → B i } i ∈ N for a II -equivalencerelation R on ( X, µ ) the cost of the graphing
Φ iscost(Φ) = X i µ ( A i ) = X i µ ( B i )and the cost of the relation iscost( R ) = inf { cost(Φ) : Φ generates R } . A generating systems Φ defines a graph structure on every R -class and cost(Φ) is half ofthe average valency of this graph over the space ( X, µ ).The cost of a II -relation takes values in [1 , ∞ ]. In the definition of the cost of a relationit is important that the relation is probability measure preserving, but ergodicity is notessential. The broader context includes relations with finite classes, such relations canvalues less than one. For instance, the orbit relation of a (non-ergodic) probability measurepreserving action of a finite group Γ y ( X, µ ) one getscost( R Γ y X ) = 1 − | Γ | . If R is the orbit relation of some (not necessarily free) action Γ y ( X, µ ) then cost( R ) ≤ rank(Γ), where the latter stands for the minimal number of generators for Γ. Indeed, anygenerating set { g , . . . , g k } for Γ gives a generating system Φ = { γ i : X → X } ki =1 for R Γ y X .Recall that the amenable II -relation R amen can be generated by (any) action of Z . Hencecost( R amen ) = 1 . The cost behaves nicely with respect to restriction:
Theorem 4.7 (Gaboriau [45]) . For a II -relation R : t · (cost( R t ) −
1) = cost( R ) − t ∈ R × + ) . The following is a key tool for computations of the cost:
Theorem 4.8 (Gaboriau [45]) . Let R be a treeable equivalence relation, and Φ be a graphingof R giving a tree structure to R -classes. Then cost( R ) = cost(Φ) . Conversely, for a relation R with cost( R ) < ∞ , if the cost is attained by some graphing Ψ then Ψ is a treeing of R . The above result (the first part) implies that for any essentially free action F n y ( X, µ )one has cost( R F n y X ) = n . This allowed Gaboriau to prove the following fact, answering along standing question: Corollary 4.9 (Gaboriau [44, 45]) . If essentially free probability measure preserving actionsof F n and F m are Orbit Equivalent then n = m . Note that F n and F m are commensurable for 2 ≤ n, m < ∞ , hence they have essen-tially free actions which are weakly isomorphic . The index of such weak isomorphism willnecessarily be n − m − , or m − n − (these free groups have P.F. - fixed price).The following powerful result of Greg Hjorth provides a link from treeable relations backto actions of free groups: Theorem 4.10 (Hjorth [59]) . Let R be a treeable equivalence relation with n = cost( R ) in { , , . . . , ∞} . Then R can be generated by an essentially free action of F n . The point of the proof is to show that a relation R which has a treeing graphing with average valency n admits a (treeing) graphing with a.e. constant valency 2 n .The behavior of the cost under passing to a subrelation of finite index is quite subtle –the following question is still open (to the best of author’s knowledge). Question 4.11 (Gaboriau) . Let Γ ′ be a subgroup of finite index in Γ , and Γ y ( X, µ ) bean essentially free p.m.p. action. Is it true that the costs of the orbit relations of Γ and Γ ′ are related by the index [Γ : Γ ′ ] : cost( R Γ ′ y X ) − ′ ] · (cost( R Γ y X ) −
1) ?The real question concerns the situation where Γ ′ y ( X, µ ) is ergodic. In general Γ ′ has at most [Γ : Γ ′ ]-many ergodic components. The extreme case where the number ofΓ ′ -ergodic components is maximal: [Γ : Γ ′ ], corresponds to Γ y ( X, µ ) being co-inductionfrom an ergodic Γ ′ -action. In this case the above formula easily holds. The real questionlies in the other extreme where Γ ′ is ergodic.Recall that the notion of the cost is analogous to the notion of rank for groups, whererank(Γ) = inf { n ∈ N : ∃ epimorphism F n → Γ } . Schreier’s theorem states that for n ∈ N any subgroup F < F n of finite index [ F n : F ] = k is itself free: F ∼ = F k ( n − . This impliesthat for any finitely generated Γ and any finite index subgroup of Γ ′ < Γ one hasrank(Γ ′ ) − ≤ [Γ : Γ ′ ] · (rank(Γ) − > Γ > . . . be a chain of subgroups of finiteindex. One defines the rank gradient (Lackenby [76]) of the chain { Γ n } as the limit of themonotonic (!) sequence: RG(Γ , { Γ n } ) = lim n →∞ rank(Γ n ) − n ] . It is an intriguing question whether (or when) is it true that RG(Γ , { Γ n } ) depends only onΓ and not on a particular chain of finite index subgroups. One should, of course, assumethat the chains in question have trivial intersection, and one might require the chains toconsists of normal subgroups in the original group. In the case of free groups RG is indeedindependent of the chain.In [1] Abert and Nikolov prove that the rank gradient of a chain of finite index subgroupsof Γ is given by the cost of a certain associated ergodic p.m.p. Γ-action. Let us formulatea special case of this relation where the chain { Γ n } consists of normal subgroups Γ n with SURVEY OF MEASURED GROUP THEORY 35 T Γ n = { } . Let K = lim ←− Γ / Γ n denote the profinite completion corresponding to the chain.The Γ-action by left translations on the compact totally disconnected group K preservesthe Haar measure m K and Γ y ( K, m K ) is a free ergodic p.m.p. action. (let us point outin passing that this action has a spectral gap, implying strong ergodicity, iff the chain hasproperty ( τ ) introduced by Lubotzky and Zimmer [78]). Theorem 4.12 (Abert-Nikolov [1]) . With the above notations:
RG(Γ , { Γ n } ) = cost( R Γ y K ) − . One direction ( ≥ ) is easy to explain. Let K n be the closure of Γ n in K . Then K n isan open normal subgroup of K of index m = [Γ : Γ n ]. Let 1 = g , g , . . . , g n ∈ Γ berepresentatives of Γ n -cosets, and h , . . . , h k generators of Γ n with k = rank(Γ n ). Considerthe graphing Φ = { φ , . . . , φ m , ψ , . . . , ψ k } , where φ i : K n → g i K n are restrictions of g i (2 ≤ i ≤ m ), and ψ j : K n → K n are restrictions of h j (1 ≤ j ≤ k ). These maps are easilyseen to generate R Γ y K , with the cost ofcost(Φ) = k · m K ( K n ) + ( m − · m K ( K n ) = k − m + 1 = 1 + rank(Γ n ) − n ] . Abert and Nikolov observed that a positive answer to Question 4.11 combined withthe above result shows that RG(Γ) is independent of the choice of a (normal) chain, andtherefore is a numeric invariant associated to any residually finite finitely generated groups.Surprisingly, this turns out to be related to a problem in the theory of compact hyperbolic3-manifolds concerning rank versus Heegard genus [76] – see [1] for the connection andfurther discussions.The above result has an application, independent of Question 4.11. Since amenablegroups have P.F. with C = 1, it follows that a finitely generated, residually finite amenablegroup Γ has sub-linear rank growth for finite index normal subgroups with trivial intersec-tion, i.e., RG(Γ) = 0 for any such chain.4.2.5. ℓ -Betti numbers. We have already mentioned the ℓ -Betti numbers β (2) i (Γ) associ-ated with a discrete group Γ and Gaboriau’s proportionality result 3.8 for Measure Equiva-lence between groups. In fact, rather than relating the ℓ -Betti numbers of groups via ME,in [48] Gaboriau- defines the notion of ℓ -Betti numbers β (2) i ( R ) for a II -equivalence relation R ;- proves that β (2) i (Γ) = β (2) i ( R Γ y X ) for essentially free ergodic action Γ y ( X, µ );- observes that β (2) i ( R t ) = t · β (2) i ( R ) for any II -relation.The definition of β (2) i ( R ) is inspired by the definition of β (2) i (Γ) by Cheeger and Gromov[19]: it uses R -action (or groupoid action) on pointed contractible simplicial complexes,corresponding complexes of Hilbert modules with R -action, and von-Neumann dimensionwith respect to the algebra M R .In [104] Roman Sauer gives a more algebraic definition of ℓ -Betti numbers of equivalencerelations as well as more general measured groupoids, using L¨uck’s notion of dimesnion.Sauer and Thom [107] develop new homological tools (including a spectral sequence forassociated to strongly normal subrelations) to study ℓ -Betti numbers for groups, relationsand measured groupoids. Outer automorphism group.
Given an equivalence relation R on ( X, µ ) define thecorresponding automorphism group as the group of self isomorphisms:Aut( R ) = { T ∈ Aut(
X, µ ) : T × T ( R ) = R (modulo null sets) } . The subgroup Inn( R ) of inner automorphisms isInn( R ) = { T ∈ Aut(
X, µ ) : ( x, T ( x )) ∈ R for a.e. x ∈ X } This is just the full group [ R ], but the above notation emphasizes the fact that it is normalin Aut( R ) and suggest to consider the outer automorphism group Out( R ) = Aut( R ) / Inn( R ) . One might think of Out( R ) as the group of all measurable permutations of the R -classes on X . Recall (Lemma 4.2) that Inn( R ) is a huge group as it acts transitively on (classes modnull sets of) measurable subsets of any given size in X . Yet the quotient Out( R ) might besmall (even finite or trivial), and can sometimes be explicitly computed. Remark 4.13.
As an abstract group H = Inn( R ) is simple, and its automorphisms comefrom automorphisms of R ; in particular Out( H ) = Out( R ). Moreover, Dye’s reconstructiontheorem states that (the isomorphism type of) R is determined by the structure of Inn( R )as an abstract group (see [70, § I.4] for proofs and further facts).Let us also note that the operation of restriction/amplification of the relation does notalter the outer automorphism group (cf. [39, Lemma 2.2]):Out( R t ) ∼ = Out( R ) ( t ∈ R × + ) . The group Aut( R ) has a natural structure of a Polish group ([53], [55]). First, recallthat if ( Y, ν ) is a finite or infinite measure Lebesgue space then Aut(
Y, ν ) is a Polish groupwith respect to the weak topology induced from the weak (=strong) operator topologyof the unitary group of L ( Y, ν ). This defines a Polish topology on Aut( R ) when the latteris viewed as acting on the infinite measure space ( R , ˜ µ ) . However, Inn( R ) is not alwaysclosed in Aut( R ), so the topology on Out( R ) might be complicated. Alexander Kechrisrecently found the following surprising connection: Theorem 4.14 (Kechris [70, Theorem 8.1]) . If Out( R ) fails to be a Polish group, then cost( R ) = 1 . Now assume that R can be presented as the orbit relation of an essentially free actionΓ y ( X, µ ), so Aut( R ) is the group of self orbit equivalences of Γ y X . The centralizerAut Γ ( X, µ ) of Γ in Aut(
X, µ ) embeds in Aut( R ), and if Γ is ICC (i.e., has Infinite ConjugacyClasses) then the quotient map Aut( R ) out −→ Out( R ) is injective on Aut Γ ( X, µ ) (cf. [53,Lemma 2.6]). So Out( R ) has a copy of Aut Γ ( X, µ ), and the latter might be very big. Forexample in the Bernoulli action Γ y ( X, µ ) = ( X , µ ) Γ , it contains Aut( X , µ ) actingdiagonally on the factors. Yet, if Γ has property (T) then Aut Γ ( X, µ ) · Inn( R Γ y X ) is openin the Polish group Aut( R Γ y X ). In this case the image of Aut Γ ( X, µ ) has finite or countable This topology coincides with the restriction to Aut( R ) of the uniform topology on Aut( X, µ ) givenby the metric d ( T, S ) = µ { x ∈ X : T ( x ) = S ( x ) } . On all of Aut( X, µ ) the uniform topology is completebut not separable; but its restriction to Aut( R ) is separable. SURVEY OF MEASURED GROUP THEORY 37 index in Out( R Γ y X ). This fact was observed by Gefter and Golodets in [55, § R Γ y X ) one looks at OE-cocycles c T : Γ × X → Γ corresponding toelements T ∈ Aut( R Γ y X ). It is not difficult to see that c T is conjugate in Γ to the identity(i.e., c T ( g, x ) = f ( gx ) − gf ( x ) for f : X → Γ) iff T is in Aut Γ ( X, µ ) · Inn( R ). Thus, startingfrom a group Γ or action Γ y X with strong rigidity properties for cocycles, one controlsOut( R Γ y X ) via Aut Γ ( X, µ ). This general scheme (somewhat implicitly) is behind the firstexample of an equivalence relation with trivial Out( R ) constructed by Gefter [52, 53]. Hereis a variant of this construction: Theorem 4.15.
Let Γ be a torsion free group with property (T), K a compact connectedLie group without outer automorphisms, and τ : Γ → K a dense imbedding. Let L < K bea closed subgroup and consider the ergodic actions Γ y ( K, m K ) and Γ y ( K/L, m
K/L ) byleft translations. Then Out( R Γ y K ) ∼ = K, Out( R Γ y K/L ) ∼ = N K ( L ) /L. In particular, taking K = PO n ( R ) and L < K to be a stabilizer of a line in R n , thespace K/L is the projective space P n − , and we getOut( R Γ y P n − ) = { } for any property (T) dense subgroup Γ < PO n ( R ) (such Γ iff n ≥
5, Zimmer [125, Theorem7]). The preceding discussion, combined with the cocycles superrigidity theorem 5.20 be-low, and an easy observation that Aut Γ ( K/L, m
K/L ) is naturally isomorphic to N K ( L ) /L ,provide a self contained sketch of the proof of the theorem.In the above statement Out( K ) is assumed to be trivial and Γ to be torsion free just tosimplify the statement. However the assumption that K is connected is essential. Indeed,the dense embedding of Γ = PSL n ( Z ) in the compact profinite group K = PSL n ( Z p ) where p is a prime, gives Out( R PSL n ( Z ) y PSL n ( Z p ) ) ∼ = PSL n ( Q p ) ⋊ Z / Z where the Z / g g tr . The inclusion ⊃ was found in[53], and the equality is proved in [39, Theorem 1.6], where many other computations ofOut( R Γ y X ) are carried out for actions of lattices in higher rank Lie groups.Finally, we recall that the recent preprint [100] of Popa and Vaes quoted above (Theo-rem 4.5) shows that an arbitrary totally disconnected lcsc group G can arise as Out( R Γ y X )for an essentially free action of a free group F ∞ .4.2.7. Cohomology.
Equivalence relations have groups of cohomology associated to themsimilar to cohomology of groups. These were introduced by Singer [114] and largely em-phasized by Feldman and Moore [33].Given, say a type II , equivalence relation R on ( X, µ ) consider R ( n ) = (cid:8) ( x , . . . , x n ) ∈ X n +1 : ( x i , x i +1 ) ∈ R (cid:9) equipped with the infinite Lebesgue measure ˜ µ ( n ) defined by˜ µ ( n ) ( A ) = Z X n ( x , . . . , x n ) : ( x , . . . , x n ) ∈ R ( n ) o dµ ( x ) . Take ( R (0) , ˜ µ ) to be ( X, µ ). Note that ( R (1) , µ (1) ) is just ( R , ˜ µ ) from § µ isassumed to be R -invariant, the above formula is invariant under permutations of x , . . . , x n .Fix a Polish Abelian group A written multiplicatively (usually A = T ). The space C n ( R , A ) of n -cochains consists of equivalence classes (modulo ˜ µ ( n ) -null sets) of measurablemaps R ( n ) → A , linked by the operators d n : C n ( R , A ) → C n +1 ( R , A ) d n ( f )( x , . . . , x n +1 ) = n +1 Y i =0 f ( x , . . . , ˆ x i , . . . , x ) ( − i . Call Z n ( R ) = Ker( d n ) the n -cocycles, and B n ( R ) = Im( d n − ) the n -coboundaries; thecohomology groups are defined by H n ( R ) = Z n ( R ) /B n ( R ). In degree n = 1 the 1-cocyclesare measurable maps c : ( R , µ ) → A such that c ( x, y ) c ( y, x ) = c ( x, z )and 1-coboundaries have the form b ( x, y ) = f ( x ) /f ( y ) for some measurable f : X → A .If A is a compact Abelian group, such as T , then C ( R , A ) is a Polish group (with respectto convergence in measure). Being closed Z ( R , A ) is a Polish group. Schmidt [109] showedthat B ( R , A ) is closed in Z ( R , A ) iff R is strongly ergodic.There are only few cases where H ( R , T ) were computed: C.C. Moore [89] constructeda relation with trivial H ( R , T ). Gefter [54] considered H ( R Γ y G , T ) for actions of prop-erty (T) group Γ densely imbedded in a semi-simple Lie group G . More recently Popaand Sasyk [99] studied H ( R Γ y X , T ) for property (T) groups Γ with Bernoulli actions( X, µ ) = ( X , µ ) Γ . In both cases H ( R Γ y X , T ) is shown to coincide with group of charac-ters Hom(Γ , T ). Higher cohomology groups remain mysterious.The fact that A is Abelian is essential to the definition of H n ( R , A ) for n >
1, but indegree n = 1 one can define H ( R , Λ) as a set for a general target group Λ . In fact, thisnotion is commonly used in this theory under the name of measurable cocycles (see § Aand 5.1 below). For the definition in terms of equivalence relations let Z ( R , Λ) denote the set of all measurable maps (mod ˜ µ -null sets) c : ( R , ˜ µ ) → Λ s.t. c ( x, z ) = c ( x, y ) c ( y, z )and let H ( R , Λ) = Z ( R , Λ) / ∼ where the equivalence ∼ between c, c ′ ∈ Z ( R , Λ) isdeclared if c ( x, y ) = f ( x ) − c ′ ( x, y ) f ( y ) for some measurable f : ( X, µ ) → Λ.If R = R Γ y X is the orbit relation of an essentially free action, then Z ( R Γ y X , Λ)coincides with the set of measurable cocycles α : Γ × X → Λ by α ( g, x ) = c ( x, gx ). Notethat Hom(Γ , Λ) / Λ maps into H ( R , Λ), via c π ( x, y ) = π ( g ) for the unique g ∈ Γ with x = gy . The point of cocycles superrigidity theorems is to show that under favorableconditions this map is surjective. In order to define the notion of measurability Λ should have a Borel structure, and better be a Polishgroup; often it is a discrete countable groups, or Lie group.
SURVEY OF MEASURED GROUP THEORY 39
Free decompositions.
Group theoretic notions such as free products, amalgamatedproducts, and HNN-extensions can be defined in the context of equivalence relations – seeGaboriau [45, Section IV]. For example, a II -relation R is said to split as a free productof sub-relations { R i } i ∈ I , denoted R = ∗ i ∈ I R i , if(1) R is generated by { R i } i ∈ I , i.e., R is the smallest equivalence relation containingthe latter family;(2) Almost every chain x = x , . . . , x n = y , where x j − = x j , ( x j − , x j ) ∈ R i ( j ) and i ( j + 1) = i ( j ), has x = y .If S is yet another subrelation, one says that R splits as a free product of R i amalga-mated over S , R = ∗ S R i , if in condition (2) one replaces x j − = x j by ( x j − , x j ) S .The obvious example of the situation above is an essentially free action of a free productof groups Γ = Γ ∗ Γ (resp. amalgamated product Γ = Γ ∗ Γ Γ ) on a probabilityspace ( X, µ ); in this case the orbit relations R i = R Γ i y X satisfy R = R ∗ R (resp. R = R ∗ R R ).Here is another useful construction (see [65]). Let R , R be measure preserving (possiblyergodic) relations on a probability space ( X, µ ). It can be shown that for a residual set of T ∈ Aut(
X, µ ) the relation generated by R and T ( R ) is their free amalgamated product.In the category of groups the isomorphism type of the free product Γ ∗ Γ depends only onisomorphism types of the free factors. This is not so in the context of relations.In [7] Alvarez and Gaboriau further study free decompositions of relations and developan analogue of Bass-Serre theory in this context. In particular, a II -relation R is definedto be freely indecomposable (FI) if R is not a free product of its subrelations. A groupΓ is said to be measurably freely indecomposable (MFI) if all its essentially free actiongive freely indecomposable orbit relations. A group may fail to be MFI even if it is freelyindecomposable in the group theoretic sense (surface groups provide an example). Notsurprisingly, groups with property (T) are MFI; but more generally Theorem 4.16 (Alvarez-Gaboriau [7]) . If Γ is non-amenable and β (2)1 (Γ) = 0 then Γ isMFI. Theorem 4.17 (Alvarez-Gaboriau [7]) . Let I , J be two finite or countable index sets, { Γ i } i ∈ I and { Λ j } j ∈ J be two families of MFI groups, Γ = ∗ i ∈ I Γ i , Λ = ∗ j ∈ J Λ j , and Γ y ( X, µ ) , Λ y ( Y, ν ) be essentially free p.m.p. actions where each Γ i y ( X, µ ) and Λ j y ( Y, ν ) are ergodic. Assume that Γ y X SOE ∼ Λ y Y .Then | I | = | J | and there is a bijection θ : I → J so that Γ i y X SOE ∼ Λ θ ( i ) y Y . The assumption that each free factor is ergodic is important here. Yet Alvarez andGaboriau analyze which statements remain true when this assumption is dropped.4.3.
Rigidity of equivalence relations.
The close relation between ME and SOE allowsto deduce that certain orbit relations R Γ y X remember the acting group Γ and the actionΓ y ( X, µ ) up to isomorphism, or up to a virtual isomorphism . This slightly technicalconcept is described in the following:
Lemma 4.18.
Suppose an ergodic ME-coupling (Ω , m ) of Γ with Λ corresponds to a SOEbetween ergodic actions T : Γ y ( X, µ ) SOE ∼ Λ y ( Y, ν ) . Then the following are equivalent: (1) There exist short exact sequences → Γ → Γ → Γ → , → Λ → Λ → Λ → where Γ and Λ are finite, a discrete (Γ , Λ ) -coupling (Ω , m ) and an equivariantmap Φ : (Ω , m ) → (Ω , m ) ;(2) There exist isomorphism between finite index subgroups Γ > Γ ∼ = Λ < Λ , so that Γ y X = X/ Γ and Λ y Y = Y / Λ are induced from some isomorphicergodic actions Γ y X ∼ = Λ y Y .(3) The SOE (or ME) cocycle Γ × X → Λ is conjugate in Λ to a cocycle whose restric-tion to some finite index subgroup Γ is a homomorphism Γ → Λ (the image isnecessarily of finite index). Let us now state two general forms of relation rigidity. Here is one form
Theorem 4.19.
Let Γ y ( X, µ ) be an ergodic essentially free action of one of the typesbelow, Λ an arbitrary group and Λ y ( Y, ν ) as essentially free p.m.p. action whose orbitrelation R Λ y Y is weakly isomorphic to R Γ y X .Then Λ is commensurable up to finite kernels to Γ and the actions Γ y X and Λ y Y are virtually isomorphic; in particular, the SEO-index is necessarily rational.The list of actions Γ y X with this SOE-rigidity property include:(1) Γ is a lattice in a connected, center free, simple, Lie group G of higher rank, and Γ y X has no equivariant quotients of the form Γ y G/ Γ ′ where Γ ′ < G is a lattice( [37, Theorem A] );(2) Γ = Γ × · · · × Γ n where n ≥ , Γ i ∈ C reg , and Γ i y ( X, µ ) are ergodic; in additionassume that Λ y ( Y, ν ) is mildly mixing (Monod-Shalom [88] );(3) Γ is a finite index subgroup in a (product of ) Mapping Class Groups as in Theo-rem 3.22 (Kida [75] ). (a) For a concrete example for (1)–(3) one might take Bernoulli actions Γ y ( X , µ ) Γ .In (1) one might also take SL n ( Z ) y T n or SL n ( Z ) y SL n ( Z p ) with n ≥
3. In (2) one mightlook at F n × F m acting on a compact Lie group K , e.g. SO ( R ), by ( g, h ) : k gkh − where F n , F m are imbedded densely in K . (b) In (1) the assumption that there are no Γ-equivariant quotient maps X → G/ Γ ′ isnecessary, since given such a quotient there is a Γ ′ -action on some ( X ′ , µ ′ ) with Γ ′ y X ′ SOE ∼ Γ y X . The rigidity statement in this case is that this is a complete list of groups andtheir essentially free actions up to virtual isomorphism ([37, Theorem C]). The appearanceof these factors has to do with ( G, m G ) appearing as a quotient of a (Γ , Λ)-coupling. (c)
The basic technique for establishing the stated rigidity in cases (1) – (3) is to establishcondition (1) in Lemma 4.18. This is done by analyzing a self Γ-coupling of the form Ω × Λ ˇΩ(where X = Ω / Λ and Y = Ω / Γ) and invoking an analogue of the construction in § (d) In all cases one can sharpen the results (eliminate the ”virtual”) by imposing somebenign additional assumptions: rule out torsion in the acting groups, and impose ergodicityfor actions of finite index subgroups.
SURVEY OF MEASURED GROUP THEORY 41
The second stronger form of relation rigidity refers to rigidity of relation morphisms which are obtained from G dsc - cocycle superrigid actions discovered by Sorin Popa (see § Theorem 4.20.
Let Γ y ( X, µ ) be a mixing G dsc -cocycle superrigid action, such as:(1) A Bernoulli Γ -action on ( X , µ ) Γ , where Γ has property (T), or Γ = Γ × Γ with Γ non-amenable and Γ being infinite;(2) Γ y K/L where Γ → K is a homomorphism with dense image in a simple compactLie group K with trivial π ( K ) , L < K is a closed subgroup, and Γ has (T).Let Λ be some group with an ergodic essentially free measure preserving action Λ y ( Y, ν ) , X ′ ⊂ X a positive measure subset and T : X ′ → Y a measurable map with T ∗ µ ≺ ν and ( x , x ) ∈ R Γ y X ∩ ( X ′ × X ′ ) = ⇒ ( T ( x ) , T ( x )) ∈ R Λ y Y . Then there exist • an exact sequence Γ −→ Γ ρ −→ Λ with finite Γ and Λ < Λ ; • A Λ -ergodic subset Y ⊂ Y with < ν ( Y ) < ∞ ; • Denoting ( X , µ ) = ( X, µ ) / Γ and ν = ν ( Y ) − · ν | Y , there is an isomorphism T : ( X , µ ) ∼ = ( Y , ν ) of Λ -actions.Moreover, µ -a.e. T ( x ) and T (Γ x ) are in the same Λ -orbit. A question of Feldman and Moore.
Feldman and Moore showed [32] that any count-able Borel equivalence relation R can be generated by a Borel action of a countable group.They asked whether one can find a free action of some group, so that R -classes would bein one-to-one correspondence with the acting group. This question was answered in thenegative by Adams [3]. In the context of measured relations, say of type II , the real ques-tion is whether it is possible to generate R (up to null sets) by an essentially free action ofsome group. This question was also settled in the negative in [37, Theorem D], using thefollowing basic constructions:(1) Start with an essentially free action Γ y ( X, µ ) which is rigid as in Theorem 4.19or 4.20, and let R = ( R Γ y X ) t with an irrational t .(2) Consider a proper imbedding G ֒ → H of higher rank simple Lie groups choose alattice Γ < H , say G = SL ( R ) ⊂ H = SL ( R ) with Γ = SL ( Z ). Such actionsalways admit a Borel cross-section X ⊂ H/ Γ for the G -action, equipped with aholonomy invariant probability measure µ . Take R on ( X, µ ) to be the relation ofbeing in the same G -orbit.In case (1) one argues as follows: if some group Λ has an essentially free action Λ y ( Y, ν )with ( R Γ y X ) t = R ∼ = R Λ y Y then the rigidity implies that Γ and Λ are commensurableup to finite kernel, and Γ y X is virtually isomorphic to Λ y Y . But this would implythat the index t is rational, contrary to the assumption. This strategy can be carried outin other cases of very rigid actions as in [62, 75, 88, 96]. Theorem 5.20 provides an exampleof this type R = ( R Γ y K ) t where Γ is a Kazhdan group densely imbedded in a compactconnected Lie group K . So the reader has a sketch of the full proof for a II -relation whichcannot be generated by an essentially free action of any group. The space (
Y, ν ) might be finite or infinite Lebesgue measure space.
Example of type (2) was introduced by Zimmer in [127], where it was proved that therelation R on such a cross-section cannot be essentially freely generated by a group Λwhich admits a linear representation with an infinite image. The linearity assumption wasremoved in [37]. This example is particularly interesting because it cannot be ”repaired”by restriction/amplification, because any R t can be realized as a cross-section of the same G -flow on H/ Γ. Question (Vershik) . Let R on ( X, µ ) be a II -relation which cannot be generated by anessentially free action of a group; and let Γ y ( X, µ ) be some action producing R . Onemay assume that the action is faithful, i.e., Γ → Aut(
X, µ ) is an embedding. What can besaid about Γ and the structure of the measurable family { Γ x } x ∈ X of the stabilizers of pointsin X ? In [102] Sorin Popa and Stefaan Vaes give an example of a II -relation R (which is arestriction of the II ∞ -relation R SL ( Z ) y R to a subset A ⊂ R of positive finite measure)which has property (T) but cannot be generated by an action (not necessarily free) of any group with property (T). SURVEY OF MEASURED GROUP THEORY 43 Techniques
Superrigidity in semi-simple Lie groups.
Superrigidity is the following amazingphenomenon discovered by G. A. Margulis (see [82] for more general statements, includingproducts of semi-simple algebraic groups over local fields, and applications such as Margulis’Arithmeticity Theorem).
Theorem 5.1 (Margulis [80]) . Let G and G ′ be (semi-)simple connected real center freeLie groups without compact factors with rk( G ) ≥ , Γ < G be an irreducible lattice and π : Γ → G ′ a homomorphism with π (Γ) not being Zariski dense in G ′ and not precompact.Then π extends to a (rational) epimorphism ¯ π : G → G ′ . This remarkable result is proved by a combination of purely ergodic-theoretic argumentsand the theory of algebraic groups. The proof does not distinguish between uniform andnon-uniform lattices. The result applies to all irreducible lattices in higher rank Lie groups,including irreducible lattices in say SL ( R ) × SL ( R ). The assumption that π (Γ) is notprecompact in G ′ is redundant if π (Γ) is Zariski dense in a real Lie group G ′ (since compactgroups over R are algebraic), but is important in general (cf. SL n ( Z ) < SL n ( Q p ) is Zariskidense but precompact).Recall that considering the transitive action G y X = G/ Γ and any topological group H there is a bijection between measurable cocycles G × G/ Γ → H modulo cocycle conjugationand homomorphisms Γ → H modulo conjugation in H (see § A.1). With this interpretationMargulis’ superrigidity Theorem 5.1 is a particular case of the following seminal result ofR. J. Zimmer:
Theorem 5.2 (Zimmer [119], see also [126]) . Let G , G ′ be semi-simple Lie group as above,in particular rk R ( G ) ≥ , let G y ( X, µ ) be an irreducible probability measure preservingaction and c : G × X → G ′ be a measurable cocycle which is Zariski dense and not compact.Then there exist a (rational) epimorphism π : G → G ′ and a measurable f : X → G ′ sothat c ( g, x ) = f ( gx ) − π ( g ) f ( x ) . In the above statement irreducibility of G y ( X, µ ) means mere ergodicity if G is asimple group, and ergodicity of the actions G i y ( X, µ ) of all the factors G i in the case ofa semi-simple group G = Q G i with n ≥ < G = Q G i in a semi-simple group the transitive action G y G/ Γ is irreducible precisely iff Γ is an irreduciblelattice in G . The notions of being Zariski dense (resp. not compact ) for a cocycle c : G × X → H mean that c is not conjugate to a cocycle c f taking values in a properalgebraic (resp. compact) subgroup of H .The setting of cocycles over p.m.p. actions adds a great deal of generality to the super-rigidity phenomena. First illustration of this is the fact that once cocycle superrigidity isknown for actions of G it passes to actions of lattices in G : given an action Γ y ( X, µ ) ofa lattice Γ < G one obtains a G -action on ¯ X = G × Γ X by acting on the first coordinate(just like the composition operation of ME-coupling § c : Γ × X → H has anatural lift to ¯ c : G × ¯ X → H and its cohomology is directly related to that of the originalcocycle. So cocycle superrigidity theorems have an almost automatic bootstrap from lcscgroups to their lattices. The induced action G y ¯ X is ergodic iff Γ y X is ergodic; how-ever irreducibility is more subtle. Yet, if Γ y ( X, µ ) is mixing then G y ¯ X is mixing andtherefore is irreducible. Theorem 3.14 was the first application of Zimmer’s cocycle superrigidity 5.2 (see [119]).Indeed, if α : Γ × X → Γ ′ is the rearrangement cocycle associated to an Orbit Equivalence T : Γ y ( X, µ ) OE ∼ Γ ′ y ( X ′ , µ ′ ) where Γ < G , Γ ′ < G ′ are lattices, then, viewing α astaking values in G ′ , Zimmer observes that α is Zariski dense using a form of Borel’s densitytheorem and deduces that G ∼ = G ′ (here for simplicity the ambient groups are assumed tobe simple, connected, center-free and rk R ( G ) ≥ π : Γ → G ′ and f : X → G ′ so that α ( γ, x ) = f ( γx ) π ( γ ) f ( x ) − with π : Γ → π (Γ) < G ′ being isomorphism of lattices. Remark 5.3.
At this point it is not clear whether π (Γ) should be (conjugate to) Γ ′ , and evenassuming π (Γ) = Γ ′ whether f takes values in Γ ′ . In fact, the self orbit equivalence of the Γaction on G/ Γ given by g Γ g − Γ gives a rearrangement cocycle c : Γ × G/ Γ → Γ whichis conjugate to the identity Γ → Γ by a unique map f : G/ Γ → G with f ∗ ( m G/ Γ ) ≺ m G .However, if π (Γ) = Γ ′ and f takes values in Γ ′ it follows that the original actions Γ y ( X, µ )and Γ ′ y ( X ′ , µ ′ ) are isomorphic via the identification π : Γ ∼ = Γ ′ . We return to this pointbelow.5.1.1. Superrigidity and ME-couplings.
Zimmer’s cocycle superrigidity theorem applied toOE or ME-cocycles (see § A.2, A.3) has a nice interpretation in terms of ME-couplings.Let G be a higher rank simple Lie group (hereafter implicitly, connected, and center free),denote by i : G → Aut( G ) the adjoint homomorphism (embedding since G is center free). Theorem 5.4 ([36, Theorem 4.1]) . Let G be a higher rank simple Lie group, Γ , Γ < G lattices, and (Ω , m ) an ergodic (Γ , Γ ) -coupling. Then there exists a unique measurablemap Φ : Ω → Aut( G ) so that m -a.e. on ΩΦ( γ ω ) = i ( γ )Φ( ω ) , Φ( γ ω ) = Φ( ω ) i ( γ ) − ( γ i ∈ Γ i ) . Moreover, Φ ∗ m is either the Haar measure on a group G ∼ = Ad( G ) ≤ G ′ ≤ Aut( G ) , or isatomic in which case Γ and Γ are commensurable.Sketch of the proof. To construct such a Φ, choose a fundamental domain X ⊂ Ω for Γ -action and look at the ME-cocycle c : Γ × X → Γ < G . Apply Zimmer’s cocycle super-rigidity theorem to find π : Γ → G and φ : X → G . Viewing G as a subgroup in Aut( G ),one may adjust π and φ : X → Aut( G ) by some α ∈ Aut( G ), so that π is the isomorphism i : Γ → Γ , to get c ( γ , x ) = φ ( γ .x ) − i ( γ ) φ ( x ) . Define Φ : Ω → Aut( G ) by Φ( γ x ) = φ ( x ) i ( γ ) − and check that it satisfies the required re-lation. To identify the measure Φ ∗ m on Aut( G ) one uses Ratner’s theorem (via classificationof Γ -ergodic finite measures on ¯ G/ Γ ). (cid:3) Theorem 3.16 is then proved using this fact with Γ = Γ plugged into the constructionin 5.5 which describes an unknown group Λ essentially as a lattice in G .Note that there are two distinct cases in Theorem 5.4: either Φ ∗ m is atomic, in whichcase (Ω , m ) has a discrete ME-coupling as a quotient, or Φ ∗ m is a Haar measure on a Liegroup. The former case leads to a virtual isomorphism between the groups and the actions(this is case (1) in Theorem 4.19); in the latter Γ y X ∼ = Ω / Γ has a quotient of the formΓ y ¯ G/ Γ (which is [37, Theorem C]). This dichotomy clarifies the situation in Remark 5.3above. SURVEY OF MEASURED GROUP THEORY 45
Superrigidity for product groups.
Let us now turn to a brief discussion to Monod-Shalom rigidity (see §§ G ′ has rank one (say G ′ = PSL ( R )), while G has higherrank. Subgroups of G ′ which are compact or not Zarsiki dense are very degenerate (haveamenable closure). The conclusion of the superrigidity Theorems 5.1 (resp. 5.2) is thateither a representation (resp. cocycle) is degenerate, or there is an epimorphism π : G → G ′ ,which is possible if and only if G is semi-simple G = Q G i , with one of the factors G i ≃ G ′ ,and π : G → G ′ factors through G → G i ≃ G ′ . In this case the given representation oflattice extends to π (resp. the cocycle is conjugate to the epimorphism π ).This particular setting was generalized by a number of authors [5, 6, 18] replacing theassumption that the target group G ′ has rank one, by more geometric notions, such as G ′ = Isom( X ) where X is a proper CAT(-1) space. Monod and Shalom take this to the nextlevel of generality, by replacing the assumption that G has higher rank by G = G × · · · × G n of n ≥ arbitrary compactly generated (in fact, just lcsc) groups. The philosophy is that n ≥ higher rank properties for such statements. Theorem 5.5 (Monod - Shalom [87]) . Let G = G × · · · × G n be a product of n ≥ lcsc groups, G y ( X, µ ) an irreducible p.m.p. action, H is hyperbolic-like group, and c : G × X → H is a non-elementary measurable cocycle.Then there is a non-elementary closed subgroup H < H , a compact normal subgroup K ⊳ H , a measurable f : X → H , and a homomorphism ρ : G i → H /K from one of thefactors G i of G , so that the conjugate cocycle c f takes values in H , and G × X → H → H /K is the homomorphism π : G pr i −→ G i ρ −→ H /K . This beautiful theorem is proved using the technology of second bounded cohomology (developed in [16, 17, 84] and applied in this setting in [83, 87]). The terms hyperbolic-like and non-elementary in this context are defined in these terms (class C reg etc).Let us now turn to the elegant way in which Monod and Shalom apply this tool toobtain ME and OE rigidity results. Suppose Γ = Γ × · · · × Γ n , n ≥
2, be a product of”hyperbolic-like” groups, and (Ω , m ) be a self ME-coupling of Γ. Assume that Γ y Ω / Γis an irreducible action. Consider a ME-cocycle Γ × X → Γ which can be viewed as acombination of n cocycles c i : Γ × X c −→ Γ pr i −→ Γ i ( i = 1 , . . . , n ) . Recalling that the source group is a product of n ≥ i are ”hyperbolic-like” Monod and Shalom apply Theorem 5.5, checkingthat the cocycles are non-elementary. The conclusion is that modulo some reductions andfinite kernels each cocycle c i is conjugate to a homomorphism ρ i : Γ j ( i ) → Γ ′ i . The crucialobservation is that since Γ i commute, the conjugations can be performed simultaneously !This basic idea is followed by an intricate analysis of the map i → j ( i ), kernels and co-kernelsof ρ i . The result, modulo some technicalities that can be cleared out by assumptions onlack of torsion, that i → j ( i ) is a permutation and ρ i are isomorphisms; the original cocycle c can be conjugate to an automorphism of Γ!This outcome can now be plugged into an analogue of Theorem 5.4, and that can beused as an input to a construction like Theorem 5.13 with G = Γ. This allows to identify unknown groups Λ Measure Equivalent to Γ = Γ × · · · × Γ n . The only delicate point is that starting from a Γ y X SOE ∼ Λ y Y and the corresponding (Γ , Λ)-coupling Ω oneneeds to look at self Γ-coupling Σ = Ω × Λ ˇΩ and apply the cocycle superrigidity result toΓ y Σ / Γ. In order to guarantee that the latter action is irreducible, Monod and Shalomrequire Γ y X to be irreducible and Λ y Y to be mildly mixing . They also show that theargument breaks without this assumption.In a joint work with Uri Bader [10] it is proposed to study higher rank superrigidityphenomena using a notion of a (generalized) Weyl group, which works well for higher ranksimple Lie groups, arbitrary products G = G × · · · × G n of n ≥ A groups(close relatives to lattices in SL ( Q p )). Theorem 5.6 (Bader - Furman [10]) . Theorem 5.5 holds for target groups from class D ea . Here D ea is a class of hyperbolic-like groups which includes many of the examples in C reg . Plugging this into Monod-Shalom machine one obtains the same results of productsof groups in class D ea .5.3. Strong rigidity for cocycles.
In the proof of Theorem 5.4 Zimmer’s cocycle super-rigidity was applied to a Measure Equivalence cocycle. This is a rather special class of cocy-cles (see § A.3). If cocycles are analogous to representations of lattices then ME-cocycles areanalogous to isomorphisms between lattices, in particular, they have an ”inverse”. Kida’swork on ME for Mapping Class Groups focuses on rigidity results for such cocycles. Weshall not attempt to explain the ingredients used in this work, but will just formulate themain technical result analogous to Theorem 5.4. Let Γ be a subgroup of finite index inΓ(Σ g,p ) ⋄ with 3 g + p − > C = C (Σ g,p ) denote its curve complex, and Aut( C ) the groupof its automorphisms; this is a countable group commensurable to Γ. Theorem 5.7 (Kida [73]) . Let (Ω , m ) be a self ME-coupling of Γ . Then there exists ameasurable map Γ × Γ -equivariant map Φ : Ω → Aut( C ) . Returning to the point that ME-cocycles are analogous to isomorphism between lattices,one might wonder whether Theorem 5.4 holds in cases where Mostow rigidity applies, specif-ically for G of rank one with PSL ( R ) excluded. In [11] this is proved for G ≃ Isom( H n R ), n ≥
3, and a restricted ME.
Theorem 5.8 (Bader - Furman - Sauer [11]) . Theorem 5.4 applies to ℓ -ME-couplings oflattices in G = SO n, ( R ) , n ≥ . The proof of this result uses homological methods ( ℓ and other completions of the usualhomology) combined with a version of Gromov-Thurston proof of Mostow rigidity (forIsom( H n R ), n ≥
3) adapted to this setting.uperrigid5.4.
Cocycle superrigid actions.
In all the previous examples the structure of the actinggroup was the sole source for (super-)rigidity. Recently Sorin Popa has discovered a sequenceof remarkable cocycle superrigidity results of a completely different nature [92–98]. Inthese results the action Γ y ( X, µ ) is the main source of the following extreme cocyclesuperrigidity phenomena.
SURVEY OF MEASURED GROUP THEORY 47
Definition 5.9.
An action Γ y X is C - cocycle superrigid , where C is some class oftopological groups, if for any Λ ∈ C any measurable cocycle c : Γ y X → Λ has the form c ( g, x ) = f ( gx ) − ρ ( g ) f ( x ) for some homomorphism ρ : Γ → Λ and a measurable f : X → Λ.Here we shall focus on the class G dsc of all countable groups; however the followingresults hold for all cocycles taking values in a broader class U fin which contains G dsc and G cpt – separable compact groups. Note that the concept of G dsc -cocycle superrigidity isunprecedentedly strong: there is no assumption on the cocycle, the assumption on thetarget group is extremely weak, the ”untwisting” takes place in the same target group. Theorem 5.10 (Popa [96]) . Let Γ be a group with property (T), Γ y ( X, µ ) = ( X , µ ) Γ be the Bernoulli action. Then Γ y X is G dsc -cocycle superrigid. In fact, the result is stronger: it suffices to assume that Γ has relative property (T) with respect to a w-normal subgroup Γ , and Γ y ( X, µ ) has a relatively weakly mixing extension Γ y ( ¯ X, ¯ µ ) which is s-malleable , while Γ y ( ¯ X, ¯ µ ) is weakly mixing. Underthese conditions Γ y ( X, µ ) is U fin -cocycle superrigid. See [96] and [40] for the relevantdefinitions and more details. We indicate the proof (of the special case above) in § rigidity and deformation properties.Bernoulli actions, and more general malleable actions (such as Gaussian actions) supplythe deformations , the rigidity comes from property (T). In the following remarkable result,Popa further relaxes property (T) assumption. Theorem 5.11 (Popa [98]) . Let Γ be a group containing a product Γ × Γ where Γ isnon-amenable, Γ is infinite, and Γ × Γ is w-normal in Γ . Then any Bernoulli action Γ y ( X, µ ) is U fin -cocycle superrigid. Weak mixing assumption played an important role in the proofs. An opposite type ofdynamics is given by isometric actions , i.e., actions Γ y K/L where
L < K are compactgroups, Γ → K a homomorphism with dense image, and Γ acts by left translations. Totallydisconnected K corresponds to profinite completion lim ←− Γ / Γ n with respect to a chain ofnormal subgroups of finite index. Isometric actions Γ y K/L with profinite K , can becalled profinite ergodic actions of Γ – these are precisely inverse limits X = lim ←− X n of transitive Γ-actions on finite spaces. Adrian Ioana found the following ”virtually G dsc -cocycle superrigidity” phenomenon for profinite actions of Kazhdan groups. Theorem 5.12 (Ioana [62]) . Let Γ y X = K/L be an ergodic profinite action. Assumethat Γ has property (T), or a relative property (T) with respect to a normal subgroup Γ which acts ergodically on X . Then any measurable cocycle c : Γ y X → Λ into a discretegroup, is conjugate to a cocycle coming from a finite quotient X → X n , i.e., c is conjugateto a cocycle induced from a homomorphism Γ n → Λ of a finite index subgroup. In § y K/L using deformation vs. rigidity technique.5.5.
Constructing representations.
In Geometric Group Theory many QI rigidity re-sults are proved using the following trick. Given a metric space X one declares self-quasi-isometries f, g : X → X to be equivalent if sup x ∈ X d ( f ( x ) , g ( x )) < ∞ . Then equivalenceclasses of q.i. form a group , denoted QI( X ). This group contains (a quotient of) Isom( X ), which can sometimes be identified within QI X using q.i. language. If Γ is a group with wellunderstood QI(Γ) and Λ is an unknown group q.i. to Γ, then one gets a homomorphism ρ : Λ → Isom(Λ) → QI(Λ) ∼ = QI(Γ)whose kernel and image can then be analyzed.Facing a similar problem in ME category, there is a difficulty in defining an analoguefor QI(Γ). However, sometimes there is a sufficiently good substitute for this: if all/someself ME-couplings of Γ have an equivariant map into some group G containing Γ, it ispossible to construct a representation of an unknown group Λ ME to Γ into G . Thiswas originally done for lattices Γ in higher rank simple Lie groups G in [36], and similaridea was implemented by Monod-Shalom in [88], and Kida in [73]. In the latter cases G is a discrete group commensurable to Γ. In [11] we have revisited this constructionwith G = Isom( H n R ) ≃ SO n, ( R ) and obtained the following streamlined general statementcontaining the aforementioned ones as special cases. The notion of strong ICC is definedin § Theorem 5.13 (Bader-Furman-Sauer [11]) . Let Γ ME ∼ Λ be countable groups, (Ω , m ) be anergodic (Γ , Λ) -coupling, G be a lcsc group and τ : Γ → G be a homomorphism.(1) Assume that the image τ (Γ) is strongly ICC in G and that the self ME-coupling Σ = Ω × Λ ˇΩ of Γ admits a measurable map Φ : Σ → G , satisfying a.e. Φ([ γ x, γ y ]) = τ ( γ ) · Φ([ x, y ]) · τ ( γ ) − ( γ , γ ∈ Γ) . Then there exists a homomorphism ρ : Λ → G and a measurable map Ψ : Ω → G so thata.e. Φ([ x, y ]) = Ψ( x ) · Ψ( y ) − , Ψ( γz ) = τ ( γ ) · Ψ( z ) , Ψ( λz ) = Ψ( z ) · ρ ( λ ) − . (2) Assume, in addition, that the homomorphism τ : Γ → G has finite kernel and discreteimage. Then the same applies to the homomorphism ρ : Λ → G .(3) Assume, in addition, that G is a semi-simple Lie group with trivial center and nocompact factors and τ (Γ) is a lattice in G . Then ρ (Λ) is also a lattice in G and thepushforward measure Ψ ∗ m is either the Haar measure on G , or is an atomic measure, inwhich case Γ and ρ (Λ) are commensurable.(4) Assume (1), (2) and that G is a discrete group and ρ (Γ) has finite index in G . Then ρ (Λ) also has finite index in G . Local rigidity for measurable cocycles.
We have mentioned in § rigidityvs. deformations approach pioneered by Sorin Popa. The source of the rigidity in mostof these results is Kazhdan’s property (T), or relative property (T), or a certain spectralgap assumption. Let us illustrate this rigidity phenomenon in the context of groups withproperty (T).One of the several equivalent forms of property (T) is the following statement: a lcscgroup G has (T) if there exist a compact K ⊂ G and ǫ > G -representation π and any ( K, ǫ )-almost invariant unit vector v there exists a G -invariantunit vector w with k v − w k < /
4. The following is a variant of Hjorth’s [58].
SURVEY OF MEASURED GROUP THEORY 49
Proposition 5.14.
Let G be a group with property (T) and ( K, ǫ ) as above. Then for anyergodic probability measure preserving action G y ( X, µ ) , any countable group Λ and anypair of cocycles α, β : G × X → Λ with µ { x ∈ X : α ( g, x ) = β ( g, x ) } > − ǫ ∀ g ∈ K ) there exists a measurable map f : X → Λ so that β = α f . Moreover, one can assume that µ { x : f ( x ) = e } > / .Proof. Let ˜ X = X × Λ be equipped with the infinite measure ˜ µ = µ × m Λ where m Λ standsfor the Haar (here counting) measure on Λ. Then G acts on ( ˜ X, ˜ µ ) by g : ( x, λ ) ( g.x, α ( g, x ) λβ ( g, x ) − ) . This action preserves ˜ µ and we denote by π the corresponding unitary G -representation on L ( ˜ X, ˜ µ ). The characteristic function v = X ×{ e } satisfies k v − π ( g ) v k = 2 − h π ( g ) v, v i < − − ǫ ǫ ( g ∈ Γ)and therefore there exists a π ( G )-invariant unit vector w ∈ L ( ˜ X, ˜Λ) with k v − w k < / k w k = R X P λ | w ( x, λ ) | we may define p ( x ) = max λ | w ( x, λ ) | , Λ( x ) = { λ : | w ( x, λ ) | = p ( x ) } and observe that p ( x ) and the cardinality k ( x ) of the finite set Λ( x ) are measurable Γ-invariant functions on ( X, µ ); hence are a.e. constants p ( x ) = p ∈ (0 , k ( x ) = k ∈{ , , . . . } . Since 1 / > k v − w k ≥ (1 − p ) we have p > /
4. It follows that k = 1because 1 = k w k ≥ kp . Therefore we can write Λ( x ) = { f ( x ) } for some measurable map f : X → Λ. The π ( G )-invariance of w gives π ( G )-invariance of the characteristic functionof n ( x, f ( x )) ∈ ˜ X : x ∈ X o , which is equivalent to(5.1) f ( gx ) = α ( g, x ) f ( x ) β ( g, x ) − and β = α f . Let A = f − ( { e } ) and a = µ ( A ). Since P λ | w ( x, λ ) | is a G -invariant function it is a.e.constant k w k = 1. Hence for x / ∈ A we have | w ( x, e ) | ≤ − | w ( x, f ( x )) | = 1 − p , and116 > k v − w k ≥ a · (1 − p ) + (1 − a ) · (1 − (1 − p )) ≥ (1 − a ) · p > − a )16 . Thus a = µ { x ∈ X : f ( x ) = e } > / > / (cid:3) Cohomology of cocycles.
Let us fix two groups Γ and Λ. There is no real assumptionon Γ, it may be any lcsc group, but we shall impose an assumption on Λ. One might focuson the case where Λ is a countable group (class G dsc ), but versions of the statements belowwould apply also to separable compact groups, or groups in a larger class U fin of all Polishgroups which imbed in the unitary group of a von-Neumann algebra with finite faithfultrace , or a potentially even larger class G binv of groups with a bi-invariant metric, and theclass G alg of connected algebraic groups over local fields, say of zero characteristic. This class, introduced by Popa contains both discrete countable groups and separable compact ones.
Given a (not necessarily free) p.m.p. action Γ y ( X, µ ) let Z ( X, Λ), or Z (Γ y X, Λ),denote the space of all measurable cocycles c : Γ × X → Λ and by H ( X, Λ), or H (Γ y X, Λ), the space of equivalence classes of cocycles up to conjugation by measurable maps f : X → Λ. If Λ ∈ G alg we shall focus on a subset H ss ( X, Λ) of (classes of) cocycles whosealgebraic hull is connected, semi-simple, center free and has no compact factors.Any Γ-equivariant quotient map π : X → Y defines a pull-back Z ( Y, Λ) → Z ( X, Λ) by c π ( g, x ) = c ( g, π ( x )), which descends to H ( Y, Λ) π ∗ −→ H ( X, Λ) . Group inclusions i : Λ < ¯Λ, and j : Γ ′ < Γ give rise to push-forward maps H ( X, Λ) i ∗ −→ H ( X, ¯Λ) , H (Γ y X, Λ) j ∗ −→ H (Γ ′ y X, Λ) . Question.
What can be said about these maps of the cohomology ?
The discussion here is inspired and informed by Popa’s [96]. In particular, the followingstatements 5.15(2), 5.16, 5.17(1), 5.19 are variations on Popa’s original [96, Lemma 2.11,Proposition 3.5, Lemma 3.6, Theorem 3.1]. Working with class G binv makes the proofs moretransparent than in U fin – this was done in [40, § Proposition 5.15.
Let π : X → Y be a Γ -equivariant quotient map. Then H ( Y, Λ) π ∗ −→ H ( X, Λ) is injective in the following cases:(1) Λ is discrete and torsion free.(2) Λ ∈ G binv and π : X → Y is relatively weakly mixing.(3) Λ ∈ G alg and H ( − , Λ) is replaced by H ss ( − , Λ) . The notion of relative weakly mixing was introduced independently by Zimmer [117]and Furstenberg [43]: a Γ-equivariant map π : X → Y is relatively weakly mixing if the Γ-action on the fibered product X × Y X is ergodic (or ergodic relatively to Y ); this turns outto be equivalent to the condition that Γ y X contains no intermediate isometric extensionsof Γ y Y . Proposition 5.16.
Let i : Λ < ¯Λ ∈ G binv be a closed subgroup, and Γ y ( X, µ ) some p.m.p.action. Then H ( X, Λ) i ∗ −→ H ( X, ¯Λ) is injective. This useful property fails in G alg setting: if Γ < G is a lattice in a (semi-) simple Liegroup and c : Γ × G/ Γ → Γ in the canonical class then viewed as cocycle into
G > Γ, c is conjugate to the identity imbedding Γ ∼ = Γ < G , but as a Γ-valued cocycle it cannot be”untwisted”. SURVEY OF MEASURED GROUP THEORY 51
Proposition 5.17.
Let π : X → Y be a quotient map of ergodic actions, and j : Γ ′ < Γ be a normal, or sub-normal, or w-normal closed subgroup acting ergodically on X . Assumethat either(1) Λ ∈ G binv and π is relatively weakly mixing, or(2) Λ ∈ G alg and one considers H ss ( − , Λ) .Then H (Γ y Y, Λ) is the push-out of the rest of the following diagram: H (Γ y X, Λ) j ∗ / / H (Γ ′ y X, Λ) H (Γ y Y, Λ) π ∗ O O j ∗ / / H (Γ ′ y Y, Λ) π ∗ O O In other words, if the restriction to Γ ′ y X of a cocycle c : Γ × X → Λ is conjugate to onedescending to Γ ′ × Y → Λ , then c has a conjugate that descends to Γ × X → Λ . The condition Γ ′ < Γ is w-normal (weakly normal) means that there exists a wellordered chain Γ i of subgroups starting from Γ ′ and ending with Γ, so that Γ i ⊳ Γ i +1 and forlimit ordinals Γ j = S i Let π i : X → Y i , ≤ i ≤ n , be a finite collection of Γ -equivariant quotients,and Z = V ni =1 Y i . Then H ( Z, Λ) is the push-out of H ( Y i , Λ) under conditions (1)-(3) ofProposition 5.15: H ( X, Λ) H ( Y , Λ) π ∗ nnnnnnnnnnnn · · · H ( Y i , Λ) · · · π ∗ i O O H ( Y n , Λ) π ∗ n h h PPPPPPPPPPPP H ( Z, Λ) p ∗ nnnnnnnnnnnn p ∗ i O O p ∗ n h h PPPPPPPPPPPP More precisely, if c i : Γ × X → Λ are cocycles (in case (3) assume [ c i ] ∈ H ss ( Y i , Λ) ), whosepullbacks c i ( g, π i ( x )) are conjugate over X , then there exists a unique class [ c ] ∈ H ( Z, Λ) ,so that c ( g, p i ( y )) ∼ c i ( g, y ) in Z ( Y i , Λ) for all ≤ i ≤ n . The proof of this Theorem relies on Proposition 5.15 and contains it as a special case n = 1.This result can be useful to push cocycles to deeper and deeper quotients; if π : X → Y is a minimal quotient to which a cocycle or a family of cocycles can descend up to conjugacy,then it is the minimal or characteristic quotient for these cocycles: if they descend toany quotient X → Y ′ then necessarily X → Y ′ → Y . For example if Γ < G is a higher rank lattice, Λ a discrete group and c : Γ × X → Λ is an OE (or ME) cocycle, then either c descends to a Γ-action on a finite set (virtual isomorphism case), or to X π −→ G/ Λ ′ withΛ ≃ Λ ′ lattice in G , where π is uniquely defined by c (initial OE or ME).An important special (and motivating) case of Theorem 5.18 is that of X = Y × Y whereΓ y Y is a weakly mixing action. Then the projections π i : X → Y i = Y , i = 1 , 2, give Z = Y ∧ Y = { pt } and H (Γ y { pt } , Λ) = Hom(Γ , Λ). So Corollary 5.19 (Popa [96, Theorem 3.1], see also [40, Theorem 3.4]) . Let Γ y Y be aweakly mixing action and c : Γ × Y → Λ a cocycle into Λ ∈ G binv . Let X = Y × Y withthe diagonal Γ -action, c , c : Γ × X → Λ the cocycles c i ( g, ( y , y )) = c ( g, y i ) . If c ∼ c over X then there exists homomorphism ρ : Γ → Λ and a measurable f : Y → Λ , so that c ( g, y ) = f ( gy ) − ρ ( g ) f ( y ) . Proofs of some results. In this section we shall give a relatively self contained proofsof some of the results mentioned above.5.8.1. Sketch of a proof for Popa’s cocycle superrigidity theorem 5.10. First note that with-out loss of generality the base space ( X , µ ) of the Bernoulli action may be assumed to benon-atomic. Indeed, Proposition 5.15(2) implies that for each of the classes G dsc ⊂ U fin ⊆ G binv the corresponding cocycle superrigidity descends through relatively weakly mixingquotients, and ([0 , , dx ) Γ → ( X , µ ) Γ is such.Given any action Γ y ( X, µ ) consider the diagonal Γ-action on ( X × X, µ × µ ) and itscentralizer Aut Γ ( X × X ) in the Polish group Aut( X × X, µ × µ ). It always contain theflip F : ( x, y ) ( y, x ). Bernoulli actions Γ y X = [0 , Γ have the property (called s-malleability by Popa) that there is a path p : [1 , → Aut Γ ( X × X ) with p = Id and p = F . Indeed, the diagonal component-wise action of Aut([0 , × [0 , X × X =([0 , × [0 , Γ embeds into Aut Γ ( X × X ) and can be used to connect Id to F .Given any cocycle c : Γ y X → Λ one has the two lifts c i : Γ y X × X → Λ , c i ( g, ( x , x )) = c ( g, x i ) , ( i = 1 , , been connected by a continuous path of cocycles c t ( g, ( x, y )) = c ( g, p t ( x, y )), 1 ≤ t ≤ c and c are conjugate over X × X , and the proof iscompleted invoking Corollary 5.19. Under the weaker assumption of relative property (T)with respect to a w-normal subgroup, Popa uses Proposition 5.17.5.8.2. A cocycle superrigidity theorem. We state and prove a cocycle superrigidity theorem,inspired and generalizing Adrian Ioana’s Theorem 5.12. Thus a number of statements(Theorems 4.15, 4.20(2), § deformation vs. rigidity approach.Recall that an ergodic p.m.p. action Γ y ( X, µ ) is said to have a discrete spectrum ifthe Koopman Γ-representation on L ( X, µ ) is a Hilbert sum of finite dimensional subrepre-sentations. Mackey proved (generalizing Halmos - von Neumann theorem for Z , and usingPeter-Weyl ideas) that discrete spectrum action is measurably isomorphic to the isometricΓ-action on ( K/L, m K/L ), g : kL τ ( g ) kL , where L < K are compact separable groupsand τ : Γ → K is a homomorphism with dense image. Theorem 5.20 (after Ioana’s Theorem 5.12, [62]) . Let Γ y ( X, µ ) be an ergodic p.m.p.action with discrete spectrum. Assume that Γ has property (T), or contains a w-normal SURVEY OF MEASURED GROUP THEORY 53 subgroup Γ with property (T) acting ergodically on ( X, µ ) . Let Λ be an arbitrary torsionfree discrete countable group and c : Γ × X → Λ be a measurable cocycle.Then there is a finite index subgroup Γ < Γ , a Γ -ergodic component X ⊂ X ( µ ( X ) =[Γ : Γ ] − ), a homomorphism ρ : Γ → Λ and map φ : X → Λ , so that the conjugatecocycle c φ restricted to Γ y X → Λ , is the homomorphism ρ : Γ → Λ . The cocycle c φ : Γ × X → Λ is induced from ρ . The assumption that Λ is torsion free is not essential; in general, one might need tolift the action to a finite cover ˆ X → X via a finite group which imbeds in Λ. If K isa connected Lie group, then Γ = Γ and X = X = K/L . The stated result is deducedfrom the case where L is trivial, i.e. X = K , using Proposition 5.15(1). We shall makethis simplification and assume Γ has property (T) (the modification for the more generalcase uses an appropriate version of Proposition 5.14 and Proposition 5.16). An appropriatemodification of the result handles compact groups as possible target group Λ for the cocycle. Proof. The K -action by right translations: t : x xt − , commutes with the Γ-action on K (in fact, K is precisely the centralizer of Γ in Aut( K, m K )). This allows us to deform theinitial cocycle c : Γ × X → Λ be setting c t ( g, x ) = c ( g, xt − ) ( t ∈ K ) . Let F ⊂ Γ and ǫ > U of e ∈ K for every t ∈ U there is a unique measurable f t : K → Λ with c t ( g, x ) = c ( g, xt − ) = f t ( gx ) c ( g, x ) f t ( x ) − µ { x : f t ( x ) = e } > . Suppose t, s ∈ U and ts ∈ U . Then f ts ( gx ) c ( g, x ) f ts ( x ) − = c ts ( g, x ) = c ( g, xs − t − )= f t ( gxs − ) c ( g, xs − ) f t ( xs − ) − = f t ( gxs − ) f s ( gx ) c ( g, x ) [ f t ( xs − ) f s ( x ) − ] − . This can be rewritten as F ( gx ) = c ( g, x ) F ( x ) c ( g, x ) − , where F ( x ) = f ts ( x ) − f t ( xs − ) f s ( x ) . Since f t , f s , f ts take value e with probability > / 4, it follows that A = F − ( { e } ) has µ ( A ) > 0. The equation implies Γ-invariance of A . Thus µ ( A ) = 1 by ergodicity. Hencewhenever t, s, ts ∈ U (5.2) f ts ( x ) = f t ( xs − ) f s ( x ) . If K is a totally disconnected group, i.e., a profinite completion of Γ as in Ioana’s Theo-rem 5.12, then U contains an open subgroup K < K . In this case one can skip the followingparagraph.In general, let V be a symmetric neighborhood of e ∈ K so that V ⊂ U , and let K = S ∞ n =1 V n . Then K is an open (hence also closed) subgroup of K (in the connectedcase K = K ). We shall extend the family { f t : K → Λ } t ∈ V to be defined for all t ∈ K while satisfying (5.2), using ”cocycle continuation” procedure akin to analytic continuation.For t, t ′ ∈ K a V - quasi-path p t → t ′ from t to t ′ is a sequences t = t , t , . . . , t n = t ′ where t i ∈ t i − V . Two V -quasi-paths from t to t ′ are V - close if they are within V -neighborhoods from each other. Two V -quasi-paths p t → t ′ and q t → t ′ are V - homotopic if there is a chain p t → t ′ = p (0) t → t ′ , . . . , p ( k ) t → t ′ = q t → t ′ of V -quasi-paths where p ( i − and p ( i ) are V -close, 1 ≤ i ≤ k .Iterating (5.2) one may continue the definition of f · from t to t ′ along a V -quasi-path;the continuation being the same for V -close quasi-paths, and therefore for V -homotopicquasi-paths as well (all from t to t ′ ). The possible ambiguity of this cocycle continuationprocedure is encoded in the homotopy group π ( V )1 ( K ) consisting of equivalence classesof V -quasi-paths from e → e modulo V -homotopy. This group is finite. In the case of aconnected Lie group K , π ( V )1 ( K ) is a quotient of π ( K ) which is finite since K , contaninga dense property (T) group, cannot have torus factors. This covers the general case as wellsince π ( V )1 ( K ) ”feels” only finitely many factors when K is written as an inverse limit ofconnected Lie groups and finite groups. Considering the continuations of f · along V -quasi-paths e → e we get a homomorphism π ( V )1 ( K ) → Λ which must be trivial since Λ wasassumed to be torsion free. Therefore we obtain a family of measurable maps f t : K → Λindexed by t ∈ K and still satisfying (5.2).Let Γ = τ − ( K ), then [Γ : Γ ] = [ K : K ] is finite index. We shall focus on therestriction c of c to Γ y K . Note that (5.2) is a cocycle equation for the simply transitiveaction K on itself. It follows by a standard argument that it is a coboundary. Indeed, fora.e. x ∈ K equation by (5.2) holds for a.e. t, s ∈ K . In particular, for a.e. t, x ∈ K ,using s = x − x , one obtains f tx − x ( x ) = f t ( x ) f x − x ( x ), giving f t ( x ) = φ ( xt − ) φ ( x ) − , where φ ( x ) = f x − x ( x ) . Equation c t = c f t translates into the fact that the cocycle c φ ( g, x ) = φ ( gx ) − c ( g, x ) φ ( x )satisfies for a.e. x, t c φ ( g, xt − ) = c φ ( g, x ) . Essentially it does not depend on the space variable, hence it is a homomorphism c φ ( g, x ) = ρ ( g ) . The fact that c φ is induced from c φ is straightforward. (cid:3) SURVEY OF MEASURED GROUP THEORY 55 Appendix A. Cocycles Let G y ( X, µ ) be a measurable, measure-preserving (sometimes just measure classpreserving) action of a topological group G on a standard Lebesgue space ( X, µ ), and H bea topological group. A Borel measurable map c : G × X → H forms a cocycle if for every g , g ∈ G for µ -a.e. x ∈ X one has c ( g g , x ) = c ( g , g .x ) · c ( g , x )If f : X → H is a measurable map and c : G × X → H is a measurable cocycle, define the f -conjugate c f of c to be c f ( g, x ) = f ( g.x ) − c ( g, x ) f ( x ) . It is straightforward to see that c f is also a cocycle. One says that c and c f are (measurably) conjugate , or cohomologous cocycles. The space of all measurable cocycles Γ × X → Λis denoted Z (Γ y X, Λ) and the space of equivalence classes by H (Γ y X, Λ).Cocycles which do not depend on the space variable: c ( g, x ) = c ( g ) are precisely homo-morphisms c : G → H . So cocycles may be viewed as generalized homomorphisms. Infact, any group action G y ( X, µ ) defines a measured groupoid G with G (0) = X , and G (1) = { ( x, gx ) : x ∈ X, g ∈ G } (see [9] for the background). In this context cocycles canbe viewed as homomorphisms G → H .If π : ( X, µ ) → ( Y, ν ) is an equivariant quotient map between Γ-actions (so π ∗ µ = ν ,and π ◦ γ = γ for γ ∈ Γ) then for any target group Λ any cocycle c : Γ × Y → Λ lifts to¯ c : Γ × X → Λ by ¯ c ( g, x ) = c ( g, π ( x )) . Moreover, if c ′ = c f ∼ c in Z (Γ y Y, Λ) then the lifts ¯ c ′ = ¯ c f ◦ π ∼ ¯ c in Z (Γ y X, Λ); so X π −→ Y induces H (Γ y X, Λ) π ⋄ ←− H (Γ y Y, Λ)Note that Hom(Γ , Λ) is Z (Γ y { pt } , Λ) and classes of cocycles on Γ × X → Λ cohomologousto homomorphisms is precisely the pull back of H (Γ y { pt } , Λ).A.1. The canonical class of a lattice, (co-)induction. Let Γ < G be a lattice in a lcscgroup. By definition the transitive G -action on X = G/ Γ has an invariant Borel regularprobability measure µ . Let F ⊂ G be a Borel fundamental domain for the right Γ-action on G (i.e. F is a Borel subset of G set which meats every coset g Γ precisely once).Fundamental domains correspond to Borel cross-section σ : G/ Γ → G of the projection G → G/ Γ. Define: c σ : G × G/ Γ → Γ , by c σ ( g, h Γ) = σ ( gh Γ) − g σ ( h Γ) . Clearly, this is a cocycle (a conjugate of the identity homomorphism G → G ), but c σ actually takes values in Γ. This cocycle is associated to a choice of the cross-section σ (equivalently, the choice of the fundamental domain); starting from another Borel cross-section σ ′ : G/ Γ → G results in a cohomologous cocycle: c σ ′ = c fσ where f : G/ Γ → Γ is defined by σ ( x ) = f ( x ) σ ′ ( x ) . Let Γ be a lattice in G . Then any action Γ y ( X, µ ) gives rise to the induced G -action(a.k.a. suspension ) on ¯ X = G × Γ X where G -acts on the first coordinate. Equivalently,¯ X = G/ Γ × X and g : ( g ′ Γ , x ) ( gg ′ Γ , c ( g, g ′ Γ) x ) where c : G × G/ Γ → Γ is in the canonical class. Here the G -invariant finite measure ¯ µ = m G/ Γ × µ is ergodic iff µ is Γ-ergodic. If α : Γ × X → H is a cocycle, the induced cocycle ¯ α : G × ¯ X → H is given by¯ α ( g, ( g ′ Γ , x )) = α ( c ( g, g ′ Γ) , x ). The cohomology of ¯ α is the same as that of α (one relatesmaps F : ¯ X → H to f : X → H by f ( x ) = F ( e Γ , x ) taking instead of e Γ a generic point in G/ Γ). In particular, ¯ α is cohomologous to a homomorpism ¯ π : G → H iff α is cohomologousto a homomorphism Γ → H ; see [126] for details.Cocycles appear quite naturally in a number of situations such as (volume preserving)smooth actions on manifolds, where choosing a measurable trivialization of the tangent bun-dle, the derivative becomes a matrix valued cocycle. We refer the reader to David Fisher’ssurvey [35] where this type of cocycles is extensively discussed in the context of Zimmer’sprogramme. Here we shall be interested in a different type of cocycles: ”rearrangement”cocycles associated to Orbit Equivalence, Measure Equivalence etc. as follows.A.2. OE-cocycles. Let Γ y ( X, µ ) and Λ y ( Y, ν ) be two measurable, measure preserv-ing, ergodic actions on probability spaces, and T : ( X, µ ) → ( Y, ν ) be an Orbit Equiva-lence. Assume that the Λ-action is essentially free , i.e., for ν -.a.e y ∈ Y , the stabilizerΛ y = { h ∈ Λ : h.y = y } is trivial. Then for every g ∈ Γ and µ -a.e. x ∈ X , the points T ( g.x ) , T ( x ) ∈ Y lie on the same Λ-orbit. Let α ( g, x ) ∈ Λ denote the (a.e. unique) elementof Λ with T ( g.x ) = α ( g, x ) .T ( x )Considering x, g.x, g ′ g.x one checks that α is actually a cocycle α : Γ × X → Λ. We shallrefer to such α as the OE-cocycle , or the rearrangement cocycle, corresponding to T .Note that for µ -a.e. x , the map α ( − , x ) : Γ → Λ is a bijection; it describes how theΓ-names of points x ′ ∈ Γ .x translate into the Λ-names of y ′ ∈ Λ .T ( x ) under the map T .The inverse map T − : ( Y, ν ) → ( X, µ ) defines an OE-cocycle β : Λ × Y → Γ which servesas an ”inverse” to α in the sense that a.e. β ( α ( g, x ) , T ( X )) = g ( g ∈ Γ) . A.3. ME-cocycles. Let (Ω , m ) be an ME-coupling of two groups Γ and Λ and let Y, X ⊂ Ωbe fundamental domains for Γ, Λ actions respectively. The natural identification Ω / Λ ∼ = X ,Λ ω Λ ω ∩ X , translates the Γ-action on Ω / Λ to Γ y X by γ : X ∋ x gα ( g, x ) x ∈ X where α ( γ, x ) is the unique element in Λ taking γx ∈ Ω into X ⊂ Ω. It is easy to see that α : Γ × X → Λ is a cocycle with respect to the above Γ-action on X which we denote bya dot γ · x to distinguish it from the Γ-action on Ω. (If Γ and Λ are lattices in G then α : Γ × G/ Λ → Λ is the restriction of the canonical cocycle G × G/ Λ → Λ). Similarly weget a cocycle β : Λ × Y → Γ. So the (Γ , Λ) ME-coupling Ω and a choice of fundamentaldomains Y ∼ = Ω / Γ, X ∼ = Ω / Λ define a pair of cocycles(A.1) α : Γ × Ω / Λ → Λ , β : Λ × Ω / Γ → ΓChanging the fundamental domains amounts to conjugating the cocycles and vise versa. Remark A.1. One can characterize ME-cocycles among all measurable cocycles α : Γ × X → Λ as discrete ones with finite covolume . 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Measured Group Theory is an area of research that studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic-theoretic proper-ties of their actions. The paper is a survey of recent developments focused on the notion of MeasureEquivalence between groups, and Orbit Equivalence between group actions. C ONTENTS 1. Introduction 2Disclaimer 2Acknowledgements 3Organization of the paper 32. Preliminary discussion and remarks 42.1. Lattices, and other countable groups 42.2. Orbit Equivalence in Ergodic Theory 72.3. Further comments on QI, ME and related topics 93. Measure Equivalence between groups 113.1. Measure Equivalence Invariants 113.2. Orbit/Measure Equivalence Rigidity 203.3. How many Orbit Structures does a given group have? 234. Measured equivalence relations 264.1. Basic definitions 264.2. Invariants of equivalence relations 284.3. Rigidity of equivalence relations 385. Techniques 425.1. Superrigidity in semi-simple Lie groups 425.2. Superrigidity for product groups 445.3. Strong rigidity for cocycles 455.4. Cocycle superrigid actions 465.5. Constructing representations 475.6. Local rigidity for measurable cocycles 485.7. Cohomology of cocycles 495.8. Proofs of some results 51Appendix A. Cocycles 55A.1. The canonical class of a lattice, (co-)induction 55A.2. OE-cocycles 56 Date : August 10, 2010.Supported by NSF Grant DMS 0604611, and BSF Grant 2004345. A.3. ME-cocycles 56References 571. I NTRODUCTION This survey concerns an area of mathematics which studies infinite countable groups usingmeasure-theoretic tools, and studies Ergodic Theory of group actions, emphasizing the impact ofgroup structure on the actions. Measured Group Theory is a particularly fitting title as it suggestsan analogy with Geometric Group Theory . The origins of Measured Group Theory go back to theseminal paper of Robert Zimmer [139], which established a deep connection between questionson Orbit Equivalence in Ergodic Theory to Margulis’ celebrated superrigidity theorem for latticesin semi-simple groups. The notion of amenable actions, introduced by Zimmer in an earlier work[138], became an indispensable tool in the field. Zimmer continued to study orbit structures ofactions of large groups in [32, 41, 140–144, 146, 147] and [135]. The monograph [146] had a partic-ularly big impact on both ergodic theorists and people studying big groups, as well as researchersin other areas, such as Operator Algebras and Descriptive Set Theory .In the recent years several new layers of results have been added to what we called MeasuredGroup Theory, and this paper aims to give an overview of the current state of the subject. Such agoal is unattainable – any survey is doomed to be partial, biased, and outdated before it appears.Nevertheless, we shall try our best, hoping to encourage further interest in this topic. The reader isalso referred to Gaboriau’s paper [58], which contains a very nice overview of some of the resultsdiscussed here, and to Shalom’s survey [133] which is even closer to the present paper (hence thesimilarity of the titles). The monographs by Kechris and Miller [81] and the forthcoming one [80]by Kechris include topics in Descriptive Set Theory related to Measured Group Theory. Readersinterested in connections to von Neumann algebras are referred to Vaes’ [136], Popa’s [114], andreferences therein.The scope of this paper is restricted to interaction of infinite Groups with Ergodic theory, leav-ing out the connections to the theory of von Neumann algebras and Descriptive Set Theory. Whenpossible, we try to indicate proofs or ideas of proofs for the stated results. In particular, we chose toinclude a proof of one cocycle superrigidity theorem 5.20, which enables a self-contained presenta-tion of a number of important results: a very rigid equivalence relation (Theorem 4.19) with trivialfundamental group and outer automorphism group (Theorem 4.15), an equivalence relation whichcannot be generated by an essentially free action of any group ( § Disclaimer. As usual, the quoted results are often presented not in the full possible generality, sothe reader should consult the original papers for full details. The responsibility for inaccuracies,misquotes and other flaws lies solely with the author of these notes. Zimmer’s cocycle superrigidity proved in [139] plays a central role in another area of research, vigorously pursuedby Zimmer and other, concerning actions of large groups on manifolds. David Fisher surveys this direction in [42] in thisvolume. SURVEY OF MEASURED GROUP THEORY 3 Acknowledgements. I would like to express my deep appreciation to Bob Zimmer for his singularcontribution to the subject. I would also like to thank Miklos Abert, Aur´elien Alvarez, Uri Bader,Damien Gaboriau, Alexander Kechris, Sorin Popa, Yehuda Shalom, and the referee for the correc-tions and comments on the earlier drafts of the paper. Organization of the paper. The paper is organized as follows: the next section is devoted to a gen-eral introduction that emphasizes the relations between Measure Equivalence, Quasi-Isometry andOrbit Equivalence in Ergodic Theory. One may choose to skip most of this, but read Definition 2.1and the following remarks. Section 3 concerns groups considered up to Measure Equivalence. Sec-tion 4 focuses on the notion of equivalence relations with orbit relations as a prime (but not only)example. In both of these sections we consider separately the invariants of the studied objects(groups and relations) and rigidity results, which pertain to questions of classification. Section 5 de-scribes the main techniques used in these theories (mostly for rigidity): a discussion of superrigidityphenomena and some of the ad hoc tools used in the subject; generalities on cocycles appear in theappendix A. ALEX FURMAN 2. P RELIMINARY DISCUSSION AND REMARKS This section contains an introduction to Measure Equivalence and related topics and contains adiscussion of this framework. Readers familiar with the subject (especially definition 2.1 and thefollowing remarks) may skip to the next section in the first reading.There are two natural entry points to Measured Group Theory, corresponding to the ergodic-theoretic and group-theoretic perspectives. Let us start from the latter.2.1. Lattices, and other countable groups. When should two infinite discrete groups be viewed as closely related? Isomorphism of abstractgroups is an obvious, maybe trivial, answer. The next degree of closeness would be commen-surability : two groups are commensurable if they contain isomorphic subgroups of finite index.This relation might be relaxed a bit further, by allowing to pass to a quotient modulo finite nor-mal subgroups. The algebraic notion of being commensurable, modulo finite kernels, can be vastlygeneralized in two directions: Measure Equivalence (Measured Group Theory) and Quasi-Isometry(Geometric Group Theory).The key notion discussed in this paper is that of Measure Equivalence of groups, introduced byGromov in [66, 0.5.E]. Definition 2.1. Two infinite discrete countable groups G , L are Measure Equivalent (abbreviatedas ME, and denoted G ME ∼ L ) if there exists an infinite measure space ( W , m ) with a measurable,measure preserving action of G × L , so that both actions G y ( W , m ) and L y ( W , m ) admit finitemeasure fundamental domains Y , X ⊂ W : W = G g ∈ G g Y = G l ∈ L l X . The space ( W , m ) is called a ( G , L ) - coupling or ME-coupling. The index of G to L in W is the ratioof the measures of the fundamental domains [ G : L ] W = m ( X ) m ( Y ) (cid:18) = meas ( W / L ) meas ( W / G ) (cid:19) . We shall motivate this definition after making a few immediate comments. (a) The index [ G : L ] W is well defined – it does not depend on the choice of the fundamentaldomains X , Y for W / L , W / G respectively, because their measures are determined by the groupactions on ( W , m ) . However, a given pair ( G , L ) might have ME-couplings with different indices(the set { [ G : L ] W } is a coset of a subgroup of R ∗ + corresponding to possible indices [ G : G ] W of self G -couplings. Here it makes sense to focus on ergodic couplings only). (b) Any ME-coupling can be decomposed into an integral over a probability space of ergodic ME-couplings, i.e., ones for which the G × L -action is ergodic. (c) Measure Equivalence is indeed an equivalence relation between groups: for any countable G the action of G × G on W = G with the counting measure m G by ( g , g ) : g g gg − provides SURVEY OF MEASURED GROUP THEORY 5 the trivial self ME-coupling, giving reflexivity; symmetry is obvious from the definition ; whiletransitivity follows from the following construction of composition , or fusion , of ME-couplings. If ( W , m ) is a ( G , G ) coupling and ( W ′ , m ′ ) is a ( G , G ) coupling, then the quotient W ′′ = W × G W ′ of W × W ′ under the diagonal action g : ( w , w ′ ) ( g w , g − w ′ ) inherits a measure m ′′ = m × G m ′ so that ( W ′′ , m ′′ ) becomes a ( G , G ) coupling structure. The indices satisfy: [ G : G ] W ′′ = [ G : G ] W · [ G : G ] W ′ . (d) The notion of ME can be extended to the broader class of all unimodular locally compactsecond countable groups: a ME-coupling of G and H is a measure space ( W , m ) with measure spaceisomorphisms i : ( G , m G ) × ( Y , n ) ∼ = ( W , m ) , j : ( H , m H ) × ( X , m ) ∼ = ( W , m ) with ( X , m ) , ( Y , n ) being finite measure spaces, so that the actions G y ( W , m ) , H y ( W , m ) givenby g : i ( g ′ , y ) i ( gg ′ , y ) , h : j ( h ′ , x ) j ( hh ′ , x ) , commute. The index is defined by [ G : H ] W = m ( X ) / n ( Y ) . (e) Measure Equivalence between countable groups can be viewed as a category , whose ob-jects are countable groups and morphisms between, say G and L , are possible ( G , L ) couplings.Composition of morphisms is the operation of composition of ME-couplings as in (c). The triv-ial ME-coupling ( G , m G ) is nothing but the identity of the object G . It is also useful to consider quotient maps F : ( W , m ) → ( W , m ) between ( G , L ) -couplings (these are 2-morphisms in thecategory), which are assumed to be G × L non-singular maps, i.e., F ∗ [ m ] ∼ m . Since preimageof a fundamental domain is a fundamental domain, it follows (under ergodicity assumption) that m ( F − ( E )) = c · m ( E ) , E ⊂ W , where 0 < c < ¥ . ME self couplings of G which have the trivial G -coupling are especially useful, their cocycles are conjugate to isomorphisms. Similarly, ( G , L ) -couplings which have a discrete coupling as a quotient, correspond to virtual isomorphisms (seeLemma 4.18). (f) Finally, one might relax the definition of quotients by considering equivariant maps F : W → W between ( G i , L i ) -couplings ( W i , m i ) with respect to homomorphisms G → G , L → L withfinite kernels and co-kernels.Gromov’s motivation for ME comes from the theory of lattices. Recall that a subgroup G ofa locally compact second countable ( lcsc for short) group G is a lattice if G is discrete in G andthe quotient space G / G carries a finite G -invariant Borel regular measure (necessarily unique up tonormalization); equivalently, if the G -action on G by left (equivalently, right) translations admits aBorel fundamental domain of finite positive Haar measure. A discrete subgroup G < G with G / G being compact is automatically a lattice. Such lattices are called uniform or cocompact ; others are non-uniform . The standard example of a non-uniform lattice is G = SL n ( Z ) in G = SL n ( R ) . Recallthat a lcsc group which admits a lattice is necessarily unimodular.A common theme in the study of lattices (say in Lie, or algebraic groups over local fields) isthat certain properties of the ambient group are inherited by its lattices. From this perspective it isdesirable to have a general framework in which lattices in the same group are considered equivalent.Measure Equivalence provides such a framework. One should formally distinguish between ( W , m ) as a ( G , L ) coupling, and the same space with the same actions asa ( L , G ) coupling; hereafter we shall denote the latter by ( ˇ W , ˇ m ) . Example 2.2 illustrates the need to do so. ALEX FURMAN Example 2.2. If G and L are lattices in the same lcsc group G , then G ME ∼ L ; the group G with theHaar measure m G is a ( G , L ) coupling where ( g , l ) : g g g l − . (In fact, G ME ∼ G ME ∼ L if ME is considered in the broader context of unimodular lcsc groups: G ×{ pt } ∼ = G × G / G ). This example also illustrates the fact that the dual ( L , G ) -coupling ˇ G is betterrelated to the original ( G , L ) -coupling G via g g − rather than the identity map.In Geometric Group Theory the basic notion of equivalence is quasi-isometry (QI). Two metricspaces ( X i , d i ) , i = , X QI ∼ X ) if there exist maps f : X → X , g : X → X , and constants M , A so that d ( f ( x ) , f ( x ′ )) < M · d ( x , x ′ ) + A ( x , x ′ ∈ X ) d ( g ( y ) , g ( y ′ )) < M · d ( y , y ′ ) + A ( y , y ′ ∈ X ) d ( g ◦ f ( x ) , x ) < A ( x ∈ X ) d ( f ◦ g ( y ) , y ) < A ( y ∈ X ) . Two finitely generated groups are QI if their Cayley graphs (with respect to some/any finite sets ofgenerators) are QI as metric spaces. It is easy to see that finitely generated groups commensurablemodulo finite groups are QI.Gromov observes that QI between groups can be characterized as Topological Equivalence (TE)defined in the the following statement. Theorem 2.3 (Gromov [66, Theorem 0.2.C ′ ]) . Two finitely generated groups G and L are quasi-isometric iff there exists a locally compact space S with a continuous action of G × L , where bothactions G y S and L y S are properly discontinuous and cocompact. The space X in the above statement is called a TE-coupling . Here is an idea for the proof.Given a TE-coupling S one obtains a quasi-isometry from any point p ∈ S by choosing f : G → L , g : L → G so that g p ∈ f ( g ) X and l p ∈ g ( l ) Y , where X , Y ⊂ S are open sets with compact closuresand S = S g ∈ G g Y = S l ∈ L l X . To construct a TE-coupling S from a quasi-isometry f : G → L ,consider the pointwise closure of the G × L -orbit of f in the space of all maps G → L where G actsby pre-composition on the domain and L by post-composition on the image. For more details seethe guided exercise in [67, p. 98].A nice instance of QI between groups is a situation where the groups admit a common geometricmodel . Here a geometric model for a finitely generated group G is a (complete) separable metricspace ( X , d ) with a properly discontinuous and cocompact action of G on X by isometries . If X isa common geometric model for G and G , then G QI ∼ X QI ∼ G . For example, fundamental groups G i = p ( M i ) of compact locally symmetric manifolds M and M with the same universal cover˜ M ∼ = ˜ M = X have X as a common geometric model. Notice that the common geometric model X itself does not serve as a TE-coupling because the actions of the two groups do not commute.However, a TE-coupling can be explicitly constructed from the group G = Isom ( X , d ) , which islocally compact (in fact, compactly generated due to finite generation assumption on G i ) secondcountable group. Indeed, the isometric actions G i y ( X , d ) define homomorphisms G i → G withfinite kernels and images being uniform lattices. Moreover, the converse is also true: if G , G admit homomorphisms with finite kernels and images being uniform lattices in the same compactlygenerated second countable group G , then they have a common geometric model – take G with a(pseudo-)metric arising from an analogue of a word metric using compact sets.Hence all uniform lattices in the same group G are QI to each other. Yet, typically, non-uniformlattices in G are not QI to uniform ones – see Farb’s survey [37] for the QI classification for latticesin semi-simple Lie groups. To summarize this discussion : the notion of Measure Equivalence is an equivalence relation be-tween countable groups, an important instance of which is given by groups which can imbedded aslattices (uniform or not) in the same lcsc group. It can be viewed as a measure-theoretic analogueof the equivalence relation of being Quasi-Isometric (for finitely generated groups), by taking Gro-mov’s Topological Equivalence point of view. An important instance of QI/TE is given by groupswhich can be imbedded as uniform lattices in the same lcsc group. In this situation one has bothME and QI. However, we should emphasize that this is merely an analogy : the notions of QI andME do not imply each other.2.2. Orbit Equivalence in Ergodic Theory. Ergodic Theory investigates dynamical systems from measure-theoretic point of view. Hereafterwe shall be interested in measurable, measure preserving group actions on a standard non-atomicprobability measure space, and will refer to such actions as probability measure preserving ( p.m.p. ).It is often convenient to assume the action to be ergodic , i.e., to require all measurable G -invariantsets to be null or co-null (that is m ( E ) = m ( X \ E ) = orbit structures of actions. Equivalence oforbit structures is captured by the following notions of Orbit Equivalence (the notion of an orbitstructure itself is discussed in § Definition 2.4. Two p.m.p. actions G y ( X , m ) and L y ( Y , n ) are orbit equivalent (abbreviatedOE, denoted G y ( X , m ) OE ∼ L y ( Y , n ) ) if there exists a measure space isomorphism T : ( X , m ) ∼ =( Y , n ) which takes G -orbits onto L -orbits. More precisely, an orbit equivalence is a Borel isomor-phism T : X ′ ∼ = Y ′ between co-null subsets X ′ ⊂ X and Y ′ ⊂ Y with T ∗ m ( E ) = m ( T − E ) = n ( E ) , E ⊂ Y ′ and T ( G . x ∩ X ′ ) = L . T ( x ) ∩ Y ′ for x ∈ X ′ .A weak OE , or Stable OE (SOE) is a Borel isomorphism T : X ′ ∼ = Y ′ between positive measure subsets X ′ ⊂ X and Y ′ ⊂ Y with T ∗ m X ′ = n Y ′ , where m X ′ = m ( X ′ ) − · m | X ′ , n Y ′ = n ( Y ′ ) − · n | Y ′ , sothat T ( G . x ∩ X ′ ) = L . T ( x ) ∩ Y ′ for all x ∈ X ′ . The index of such SOE-map T is m ( Y ′ ) / n ( X ′ ) .In the study of orbit structure of dynamical systems in the topological or smooth category, oneoften looks at such objects as fixed or periodic points/orbits. Despite the important role these no-tions play in the underlying dynamical system, periodic orbits have zero measure and therefore areinvisible from the purely measure-theoretic standpoint. Hence OE in Ergodic Theory is a study ofthe global orbit structure. This point of view is consistent with the general philosophy of ”non-commutative measure theory”, i.e. von Neumann algebras. Specifically OE in Ergodic Theory isclosely related to the theory of II factors as follows. ALEX FURMAN In the 1940s Murray and von Neumann introduced the so called ”group-measure space” con-struction to provide interesting examples of von-Neumann factors : given a probability measurepreserving (or more generally, non-singular) group action G y ( X , m ) the associated von-Neumannalgebra M G y X is a cross-product of G with the Abelian algebra L ¥ ( X , m ) , namely the weak closure inbounded operators on L ( G × X ) of the algebra generated by the operators { f ( g , x ) f ( g g , g . x ) : g ∈ G } and { f ( g , x ) f ( x ) f ( g , x ) : f ∈ L ¥ ( X , m ) } . Ergodicity of G y ( X , m ) is equivalent to M G y X be-ing a factor. It turns out that (for essentially free ) OE actions G y X OE ∼ L y Y the associated algebrasare isomorphic M G y X ∼ = M L y Y , with the isomorphism identifying the Abelian subalgebras L ¥ ( X ) and L ¥ ( Y ) . The converse is also true (one has to specify, in addition, an element in H ( G y X , T ) )– see Feldman-Moore [39, 40]. So Orbit Equivalence of (essentially free p.m.p. group actions) fitsinto the study of II factors M G y X with a special focus on the so called Cartan subalgebra given by L ¥ ( X , m ) . We refer the reader to Popa’s 2006 ICM lecture [114] and Vaes’ Seminar Bourbaki paper[136] for some more recent reports on this rapidly developing area.The above mentioned assumption of essential freeness of an action G y ( X , m ) means that, up toa null set, the action is free; equivalently, for m -a.e. x ∈ X the stabilizer { g ∈ G : g . x = x } is trivial.This is a natural assumption, when one wants the acting group G to ”fully reveal itself” in a.e. orbitof the action. Let us now link the notions of OE and ME. Theorem 2.5. Two countable groups G and L are Measure Equivalent iff they admit essentiallyfree (ergodic) probability measure preserving actions G y ( X , m ) and L y ( Y , n ) which are StablyOrbit Equivalent. (SOE) = ⇒ (ME) direction is more transparent in the special case of Orbit Equivalence, i.e., indexone. Let a : G × X → L be the cocycle associated to an orbit equivalence T : ( X , m ) → ( Y , n ) definedby T ( g . x ) = a ( g , x ) . T ( x ) (here freeness of L y Y is used). Consider ( W , m ) = ( X × L , m × m L ) withthe actions(2.1) g : ( x , h ) ( gx , a ( g , x ) h ) , h : ( x , k ) ( x , hk − ) ( g ∈ G , h ∈ L ) . Then X × { } is a common fundamental domain for both actions (note that here freeness of G y X is used). Of course, the same coupling ( W , m ) can be viewed as ( Y × G , n × m G ) with the L -actiondefined using b : L × Y → G given by T − ( h . y ) = b ( h , y ) . T − ( y ) . In the more general settingof Stable OE one needs to adjust the definition for the cocycles (see [45]) to carry out a similarconstruction.Alternative packaging for the (OE) = ⇒ (ME) argument uses the language of equivalence rela-tions (see § Y with X via T − , one views R L y Y and R G y X as a single relation R .Taking W = R equipped with the measure ˜ m (4.1) consider the actions g : ( x , y ) ( g . x , y ) , h : ( x , y ) ( x , h . y ) ( g ∈ G , h ∈ L ) . Here the diagonal embedding X R , x ( x , x ) , gives the fundamental domain for both actions.(ME) = ⇒ (SOE). Given an ergodic ( G , L ) coupling ( W , m ) , let X , Y ⊂ W be fundamental domainsfor the L , G actions; these may be chosen so that m ( X ∩ Y ) > 0. The finite measure preservingactions(2.2) G y X ∼ = W / L , L y Y ∼ = W / G . von Neumann algebras whose center consists only of scalars. SURVEY OF MEASURED GROUP THEORY 9 have weakly isomorphic orbit relations, since they appear as the restrictions to X and Y of therelation R G × L y W (of type II ¥ ); these restrictions coincide on X ∩ Y . The index of this SOE coincideswith the ME-index [ G : L ] W (if [ G : L ] W = X = Y ).The only remaining issue is that the actions G y X ∼ = W / L , L y Y ∼ = W / G may not be essentialfree. This can be fixed (see [56]) by passing to an extension F : ( ¯ W , ¯ m ) → ( W , m ) where G y ¯ W / L and L y ¯ W / G are essentially free. Indeed, take ¯ W = W × Z × W , where L y Z and L y W are freeprobability measure preserving actions and let g : ( w , z , w ) ( g w , gz , w ) , h : ( w , z , w ) ( h w , z , hw ) ( g ∈ G , h ∈ L ) . Remark 2.6. Freeness of actions is mostly used in order to define the rearrangement cocycles fora (stable) orbit equivalence between actions. However, if SOE comes from a ME-coupling the welldefined ME-cocycles satisfy the desired rearrangement property (such as T ( g . x ) = a ( g , x ) . T ( x ) )and freeness becomes superfluous.If F : ¯ W → W is as above, and ¯ X , ¯ Y denote the preimages of X , Y , then ¯ X , ¯ Y are L , G fundamentaldomains, the OE-cocycles G y ¯ X SOE ∼ L y ¯ Y coincide with the ME-cocycles associated with X , Y ⊂ W .Another, essentially equivalent, point of view is that ME-coupling defines a weak isomorphism between the groupoids G y W / L and L y W / G . In case of free actions these groupoids reduceto their relations groupoids , but in general the information about stabilizers is carried by the ME-cocycles.2.3. Further comments on QI, ME and related topics. Let S be Gromov’s Topological Equivalence between G and L . Then any point x ∈ S defines aquasi-isometry q x : G → L (see the sketch of proof of Theorem 2.3). In ME the maps a ( − , x ) : G → L defined for a.e. x ∈ X play a similar role. However due to their measure-theoretic nature, suchmaps are insignificant taken individually, and are studied as a measured family with the additionalstructure given by the cocycle equation.Topological and Measure Equivalences are related to the following interesting notion, introducedby Nicolas Monod in [98] under the appealing term randomorphisms . Consider the Polish space L G of all maps f : G → L with the product uniform topology, and let [ G , L ] = { f : G → L : f ( e G ) = e L } . Then G acts on [ G , L ] by g : f ( x ) f ( xg ) f ( g ) − , x ∈ G . The basic observation is that homomor-phisms G → L are precisely G -fixed points of this action. Definition 2.7. A randomorphism is a G -invariant probability measure on [ G , L ] .A measurable cocycle c : G × X → L over a p.m.p. action G y ( X , m ) defines a randomorphismby pushing forward the measure m by the cocycle x c ( − , x ) . Thus Orbit Equivalence cocycles(see A.2) correspond to randomorphisms supported on bijections in [ G , L ] . Also note that the naturalcomposition operation for randomorphisms, given by the push-forward of the measures under thenatural map [ G , G ] × [ G , G ] → [ G , G ] , ( f , g ) g ◦ f corresponds to composition of couplings. The view point of topological dynamics of the G -actionon [ G , L ] may be related to quasi-isometries and Topological Equivalence. For example, points in [ G , L ] with precompact G -orbits correspond to Lipschitz embeddings G → L .2.3.1. Using ME for QI. Although Measure Equivalence and Quasi Isometry are parallel in many ways, these concepts aredifferent and neither one implies the other. Yet, Yehuda Shalom has shown [132] how one can useME ideas to study QI of amenable groups. The basic observation is that a topological coupling S of amenable groups G and L carries a G × L -invariant measure m (coming from a G -invariant prob-ability measure on S / L ), which gives a measure equivalence. It can be thought of as an invariantdistribution on quasi-isometries G → L , and can be used to induce unitary representations and coho-mology with unitary coefficients etc. from L to G . Using such constructions, Shalom [132] was ableto obtain a list of new QI invariants in the class of amenable groups, such as (co)-homology over Q ,ordinary Betti numbers b i ( G ) among nilpotent groups and others. Shalom also studied the notionof uniform embedding (UE) between groups and obtained group invariants which are monotonic with respect to UE.In [125] Roman Sauer obtains further QI-invariants and UE-monotonic invariants using a com-bination of QI, ME and homological methods.In another work [126] Sauer used ME point of view to attack problems of purely topologicalnature, related to the work of Gromov.2.3.2. ℓ p -Measure Equivalence. Let G and L be finitely generated groups, equipped with some word metrics | · | G , | · | L . We saythat a ( G , L ) coupling ( W , m ) is ℓ p for some 1 ≤ p ≤ ¥ if there exist fundamental domains X , Y ⊂ W so that the associated ME-cocycles (see A.3) a : G × X → L and b : L × Y → G satisfy ∀ g ∈ G : | a ( g , − ) | L ∈ L p ( X , m ) , ∀ h ∈ L : | b ( h , − ) | G ∈ L p ( Y , n ) . If an ℓ p -ME-coupling exists, say that G and L are ℓ p -ME. Clearly any ℓ p -ME-coupling is ℓ q forall q ≤ p . So ℓ -ME is the weakest and ℓ ¥ -ME is the most stringent among these relations. Onecan check that ℓ p -ME is an equivalence relation on groups (the ℓ p condition is preserved undercomposition of couplings), so we obtain a hierarchy of ℓ p -ME categories with ℓ -ME being theweakest (largest classes) and at p = ¥ one arrives at ME+QI. Thus ℓ p -ME amounts to MeasureEquivalence with some geometric flavor.The setting of ℓ -ME is considered in [13, 14] by Uri Bader, Roman Sauer and the author toanalyze rigidity of the least rigid family of lattices – lattices in SO n , ( R ) ≃ Isom ( H n R ) , n ≥ 3, andfundamental groups of general negatively curved manifolds. It should be noted, that examples ofnon-amenable ME groups which are not ℓ -ME seem to be rare (surface groups and free groupsseem to be the main culprits). In particular, it follows from Shalom’s computations in [131] thatfor n ≥ n , ( R ) are mutually ℓ -ME. We shall return to invariants and rigidity in ℓ -ME framework in § § SURVEY OF MEASURED GROUP THEORY 11 3. M EASURE E QUIVALENCE BETWEEN GROUPS This section is concerned with the notion of Measure Equivalence between countable groups G ME ∼ L (Definition 2.1). First recall the following deep result (extending previous work of Dye[33, 34] on some amenable groups, and followed by Connes-Feldman-Weiss [27] concerning allnon-singular actions of all amenable groups) Theorem 3.1 (Ornstein-Weiss [105]) . Any two ergodic probability measure preserving actions ofany two infinite countable amenable groups are Orbit Equivalent. This result implies that all infinite countable amenable groups are ME; moreover for any twoinfinite amenable groups G and L there exists an ergodic ME-coupling W with index [ G : L ] W = G OE ∼ L ). Measure Equivalence of all amenable groupsshows that many QI-invariants are not ME-invariants; these include: growth type, being virtuallynilpotent, (virtual) cohomological dimension, finite generations/presentation etc.The following are basic constructions and examples of Measure Equivalent groups:(1) If G and L can be embedded as lattices in the same lcsc group, then G ME ∼ L .(2) If G i ME ∼ L i for i = , . . . , n then G × · · · × G n ME ∼ L × · · · × L n .(3) If G i OE ∼ L i for i ∈ I (i.e. the groups admit an ergodic ME-coupling with index one) then ( ∗ i ∈ I G i ) OE ∼ ( ∗ i ∈ I L i ) .For 2 = n , m < ¥ the free groups F n and F m are commensurable, and therefore are ME (however, F ¥ ME ∼ F ). The Measure Equivalence class ME ( F ≤ n < ¥ ) is very rich and remains mysterious (see[58]). For example it includes: surface groups p ( S g ) , g ≥ 2, non uniform (infinitely generated)lattices in SL ( F p [[ X ]]) , the automorphism group of a regular tree, free products ∗ ni = A i of arbi-trary infinite amenable groups, more complicated free products such as F ∗ p ( S g ) ∗ Q , etc. In theaforementioned paper by Gaboriau he constructs interesting geometric examples of the form ∗ nc F g ,which are fundamental groups of certain ”branched surfaces”. Bridson, Tweedale and Wilton [19]prove that a large class of limit groups , namely all elementarily free groups, are ME to F . Noticethat ME ( F ≤ n < ¥ ) contains uncountably many groups.The fact that some ME classes are so rich and complicated should emphasize the impressive listof ME invariants and rigidity results below.3.1. Measure Equivalence Invariants. By ME- invariants we mean properties of groups which are preserved under Measure Equiva-lence, and numerical invariants which are preserved or predictably transformed as a function of theME index. The appearance of the sharper condition OE ∼ in (2) is analogous to the one in the QI context: if groups G i and L i are bi-Lipschitz then ∗ i ∈ I G i QI ∼ ∗ i ∈ I L i . Amenability, Kazhdan’s property (T), a-T-menability. These properties are defined using the language of unitary representations. Let p : G → U ( H ) bea unitary representation of a (topological) group. Given a finite (resp. compact) subset K ⊂ G and e > 0, we say that a unit vector v ∈ H is ( K , e ) - almost invariant if k v − p ( g ) v k < e for all g ∈ K .A unitary G -representation p which has ( K , e ) -almost invariant vectors for all K ⊂ G and e > weakly contain the trivial representation G , denoted G ≺ p . The trivial representation G is (strongly) contained in p , denoted G < p , if there exist non-zero p ( G ) -invariant vectors, i.e., H p ( G ) = { } . Of course G < p trivially implies G ≺ p . We recall:A MENABILITY : G is amenable if the trivial representation is weakly contained in the regularrepresentation r : G → U ( ℓ ( G )) , r ( g ) f ( x ) = f ( g − x ) .P ROPERTY (T): G has property (T) (Kazhdan [79]) if for every unitary G -representation p : G ≺ p implies G < p . This is equivalent to an existence of a compact K ⊂ G and e > G -representation p with ( K , e ) -almost invariant vectors, has non-trivialinvariant vectors. For compactly generated groups, another equivalent characterization (De-lorme and Guichardet) is that any affine isometric G -action on a Hilbert space has a fixedpoint, i.e., if H ( G , p ) = { } for any (orthogonal) G -representation p . We refer to [17] forthe details.(HAP): G is a-T-menable (or has Haagerup Approximation Property ) if the followingequivalent conditions hold: (i) G has a mixing G -representation weakly containing the triv-ial one, or (ii) G has a proper affine isometric action on a (real) Hilbert space. The classof infinite a-T-menable groups contains amenable groups, free groups but is disjoint frominfinite groups with property (T). See [24] as a reference.Measure Equivalence allows to relate unitary representations of one group to another. More con-cretely, let ( W , m ) be a ( G , L ) coupling, and p : L → U ( H ) be a unitary L -representation. Denoteby ˜ H the Hilbert space consisting of equivalence classes (mod null sets) of all measurable, L -equivariant maps W → H with square-integrable norm over a L -fundamental domain:˜ H = (cid:26) f : W → H : f ( l x ) = p ( l ) f ( x ) , Z W / L k f k < ¥ (cid:27) mod null sets . The action of G on such functions by translation of the argument, defines a unitary G -representation˜ p : G → U ( ˜ H ) . This representation is said to be induced from p : L → U ( H ) via W . (In exam-ple 2.2 this is precisely the usual Mackey induction of a unitary representations of a lattice to theambient group, followed by a restriction to another lattice).The ME invariance of the properties above (amenability, property (T), Haagerup approximationproperty) can be deduced from the following observations. Let ( W , m ) be a ( G , L ) ME-coupling, p : L → U ( H ) a unitary representation and ˜ p : G → U ( ˜ H ) the corresponding induced representation.Then:(1) If p is the regular L -representation on H = ℓ ( L ) , then ˜ p on ˜ H can be identified with the G -representation on L ( W , m ) ∼ = n · ℓ ( G ) , where n = dim L ( W / L ) ∈ { , , . . . , ¥ } .(2) If L ≺ p then G ≺ ˜ p .(3) If ( W , m ) is G × L ergodic and p is weakly mixing (i.e. 1 L < p ⊗ p ∗ ) then G < ˜ p .(4) If ( W , m ) is G × L ergodic and p is mixing (i.e. for all v ∈ H : h p ( h ) v , v i → h → ¥ in L ) then ˜ p is a mixing G -representation. SURVEY OF MEASURED GROUP THEORY 13 Combining (1) and (2) we obtain that being amenable is an ME-invariant. The deep result ofOrnstein-Weiss [105] and Theorem 2.5 imply that any two infinite countable amenable groups areME. This gives: Corollary 3.2. The Measure Equivalence class of Z is the class of all infinite countable amenablegroups ME ( Z ) = Amen . Bachir Bekka and Alain Valette [16] showed that if L does not have property (T) then it admitsa weakly mixing representation p weakly containing the trivial one. By (2) and (3) this impliesthat property (T) is a ME-invariant (this is the argument in [44, Corollary 1.4], see also Zimmer[146, Theorem 9.1.7 (b)]). The ME-invariance of amenability and Kazhdan’s property for groupsindicates that it should be possible to define these properties for equivalence relations and thenrelate them to groups. This was done by Zimmer in [138, 141] and was recently further studiedin the context of measured groupoids in [8, 9]. We return to this discussion in § Cost of groups. The notion of the cost of an action/relation was introduced by Levitt [90] and developed byDamien Gaboriau [53, 54, 56]; the monographs [81] and [80] also contain an extensive discussionof this topic.The cost of an essentially free p.m.p. action G y ( X , m ) , denoted cost ( G y X ) , is the costof the corresponding orbit relations cost ( R G y X ) as defined in § C ∗ ( G ) = inf X cost ( G y X ) , C ∗ ( G ) = sup X cost ( G y X ) where the infimum/supremum are taken over all essentially free p.m.p. actions of G (we drop er-godicity assumption here; in the definition of C ∗ ( G ) essential freeness is also superfluous). Groups G for which C ∗ ( G ) = C ∗ ( G ) are said to have fixed price, or prix fixe (abbreviated P.F.). For generalgroups, Gaboriau defined the cost of a group to be the lower one: C ( G ) = C ∗ ( G ) . To avoid confusion, we shall use here the notation C ∗ ( G ) for general groups, and reserve C ( G ) forP.F. groups only. Question 3.3. Do all countable groups have property P.F.? The importance of this question will be illustrated in § C ∗ = 1, 1 < C ∗ < ¥ , and C ∗ = ¥ are ME-invariants. More precisely: Theorem 3.4. If G ME ∼ L then C ∗ ( L ) − = [ G : L ] W · ( C ∗ ( G ) − ) for some/any ( G , L ) -coupling W . We do not know whether the same holds for C ∗ . Note that in [54] this ME-invariance is statedfor P.F. groups only. Proof. Let W be a ( G , L ) -coupling with G y X = W / L and L y Y = W / G being free, where X , Y ⊂ W are L − , G − fundamental domains. Given any essentially free p.m.p. action L y Z , consider the ( G , L ) -coupling ¯ W = W × Z with the actions g : ( w , z ) ( g w , z ) , h : ( w , z ) ( h w , hz ) ( g ∈ G , h ∈ L ) . The actions G y ¯ X = ¯ W / L and L y ¯ Y = ¯ W / G are Stably Orbit Equivalent with index [ G : L ] ¯ W =[ G : L ] W = c . Hence (using Theorem 4.7 below) we have c · ( cost ( R G y ¯ X ) − ) = cost ( R L y ¯ Y ) − . While G y ¯ X is a skew-product over G y X , the action L y ¯ Y is the diagonal action on ¯ Y = Y × Z .Since ¯ Y = Y × Z has Z as a L -equivariant quotient, it follows (by considering preimages of any”graphing system”) that cost ( L y ¯ Y ) ≤ cost ( L y Z ) . Since L y Z was arbitrary, we deduce C ∗ ( L ) − ≥ c · ( C ∗ ( G ) − ) . A symmetric argument com-pletes the proof. (cid:3) Theorem 3.5 (Gaboriau [53, 54, 56]) . The following classes of groups have P.F.:(1) Any finite group G has C ∗ ( G ) = C ∗ ( G ) = − | G | .(2) Infinite amenable groups have C ∗ ( G ) = C ∗ ( G ) = .(3) Free group F n , ≤ n ≤ ¥ , have C ∗ ( F n ) = C ∗ ( F n ) = n.(4) Surface groups G = p ( S g ) where S g is a closed orientable surface of genus g ≥ have C ∗ ( G ) = C ∗ ( G ) = g − .(5) Amalgamated products G = A ∗ C B of finite groups have P.F. with C ∗ ( G ) = C ∗ ( G ) = − ( | A | + | B | − | C | ) . In particular C ∗ ( SL ( Z )) = C ∗ ( SL ( Z )) = + .(6) Assume G , G have P.F. then the free product G ∗ G , and more general amalgamated freeproducts L = G ∗ A G over an amenable group A, has P.F. with C ( G ∗ G ) = C ( G ) + C ( G ) , C ( G ∗ A G ) = C ( G ) + C ( G ) − C ( A ) . (7) Products G = G × G of infinite non-torsion groups have C ∗ ( G ) = C ∗ ( G ) = .(8) Finitely generated groups G containing an infinite amenable normal subgroup have C ∗ ( G ) = C ∗ ( G ) = .(9) Arithmetic lattices G of higher Q -rank (e.g. SL n ≥ ( Z ) ) have C ∗ ( G ) = C ∗ ( G ) = . Note that for an infinite group C ∗ ( G ) = G has P.F. of cost one. So the content of cases (2),(7), (8), (9) is that C ∗ ( G ) = Question 3.6. Is it true that for all (irreducible) lattices G in a (semi-)simple Lie group G of higherrank have P.F. of C ∗ ( G ) = ?Is it true that any infinite group G with Kazhdan’s property (T) has P.F. with C ∗ ( G ) = ? SURVEY OF MEASURED GROUP THEORY 15 Item (9) in Theorem 3.5 provides a positive answer to the first question for some non-uniformlattices in higher rank Lie groups, but the proof relies on the internal structure of such lattices (chainsof pairwise commuting elements), rather than on its relation to the ambient Lie group G (which alsohas a lot of commuting elements). Note also that Theorem 3.4 implies that C ∗ ( G ) = ℓ -Betti number, b ( ) ( G ) = 0; while for infinite groups it was shown by Gaboriau that(3.1) b ( ) ( G ) = b ( ) ( R G y X ) ≤ cost ( R G y X ) − . Furthermore, there are no known examples of strict inequality. Lattices G in higher rank semi-simple Lie groups without property (T) still satisfy b ( ) ( G ) = b ( ) for ME groups are positively proportional by Gaboriau’s Theorem 3.8,an irreducible lattice in a product is ME to a product of lattices and products of infinite groups have b ( ) = G be agroup generated by a finite set { g , . . . , g n } and containing an infinite normal amenable subgroup A and G y ( X , m ) be an essentially free (ergodic) p.m.p. action. Since A is amenable, there is a Z -action on X with R A y X = R Z y X (mod null sets), and we let f : X → X denote the action of thegenerator of Z . Given e > E ⊂ X with 0 < m ( E ) < e so that S a ∈ A aE = S f n E = X mod null sets (if A -action is ergodic any positive measure set works; in general, oneuses the ergodic decomposition). For i = , . . . , n let f i be the restriction of g i to E . Now oneeasily checks that the normality assumption implies that F = { f , f , . . . , f n } generates R G y X , whilecost ( F ) = + n e .For general (not necessarily P.F.) groups G i a version of (6) still holds: C ∗ ( G ∗ G ) = C ∗ ( G ) + C ∗ ( G ) , C ∗ ( G ∗ A G ) = C ∗ ( G ) + C ∗ ( G ) − C ( A ) where A is finite or, more generally, amenable.Very recently Miklos Abert and Benjamin Weiss [2] showed: Theorem 3.7 (Abert-Weiss [2]) . For any discrete countable group G , the highest cost C ∗ ( G ) isattained by non-trivial Bernoulli actions G y ( X , m ) G and their essentially free quotients. Some comments are in order. Kechris [80] introduced the following notion: for probabilitymeasure preserving actions of a fixed group G say that G y ( X , m ) weakly contains G y Y ifgiven any finite measurable partition Y = F ni = Y i , a finite set F ⊂ G and an e > 0, there is a finitemeasurable partition X = F ni = X i so that (cid:12)(cid:12) m ( gX i ∩ X j ) − n ( gY i ∩ Y j ) (cid:12)(cid:12) < e ( ≤ i , j ≤ n , g ∈ F ) . The motivation for the terminology is the fact that weak containment of actions implies (but notequivalent to) weak containment of the corresponding unitary representations: L ( Y ) (cid:22) L ( X ) . Itis clear that a quotient is (weakly) contained in the larger action. It is also easy to see that the costof a quotient action is no less than that of the original (because one can lift any graphing from aquotient to the larger action maintaining the cost of the graphing). Kechris [80] proves that this(anti-)monotonicity still holds in the context of weak containment of essentially free actions offinitely generated groups, namely: G y Y (cid:22) G y X = ⇒ cost ( G y Y ) ≥ cost ( G y X ) . In fact, it follows from the more general fact that cost is upper semi-continuous in the topology ofactions. Abert and Weiss prove that Bernoulli actions (and their quotients) are weakly contained inany essentially free action of a group. Thus Theorem 3.7 follows from the monotonicity of the cost.3.1.3. ℓ -Betti numbers. The ℓ -Betti numbers of (coverings of ) manifolds were introduced by Atiyah in [11]. Cheegerand Gromov [23] defined ℓ -Betti numbers b ( ) i ( G ) ∈ [ , ¥ ] , i ∈ N , for arbitrary countable group G as dimensions (in the sense of Murray von-Neumann) of certain homology groups (which areHilbert G -modules). For reference we suggest [35], [92]. Here let us just point out the followingfacts:(1) If G is infinite amenable, then b ( ) i ( G ) = i ∈ N ;(2) For free groups b ( ) ( F n ) = n − b ( ) i ( F n ) = i > b ( ) ( G ) = b ( ) k ( G × G ) = (cid:229) i + j = k b ( ) i ( G ) · b ( ) j ( G ) .(5) Kazhdan’s conjecture, proved by L ¨uck, states that for residually finite groups satisfying ap-propriate finiteness properties (e.g. finite K ( pi , ) ) the ℓ -Betti numbers are the stable limitof Betti numbers of finite index subgroups normalized by the index: b ( ) i ( G ) = lim b i ( G n )[ G : G n ] where G > G > . . . is a chain of normal subgroups of finite index.(6) The ℓ Euler characteristic c ( ) ( G ) = (cid:229) ( − ) i · b ( ) i ( G ) coincides with the usual Euler char-acteristic c ( G ) = (cid:229) ( − ) i · b i ( G ) , provided both are defined, as is the case for fundamentalgroup G = p ( M ) of a compact aspherical manifold.(7) According to the Hopf-Singer conjecture the ℓ -Betti numbers for a fundamental group G = p ( M ) of a compact aspherical manifold M , vanish except, possibly, in the middledimension n . Atiyah’s conjecture states that ℓ -Betti numbers are integers.The following remarkable result of Damien Gaboriau states that these intricate numeric invariantsof groups are preserved under Measure Equivalence, after a rescaling by the coupling index. Theorem 3.8 (Gaboriau [55], [57]) . Let G ME ∼ L be ME countable groups. Then b ( ) i ( L ) = c · b ( ) i ( G ) ( i ∈ N ) where c = [ G : L ] W is a/the index of some/any ( G , L ) -coupling. In fact, Gaboriau introduced the notion of ℓ -Betti numbers for II -relations and related them to ℓ -Betti numbers of groups in case of the orbit relation for an essentially free p.m.p. action – seemore comments in § ℓ -Betti numbers for fundamental groups of as-pherical manifolds, such as Euler characteristic and sometimes the dimension, pass through Mea-sure Equivalence. In particular, if lattices G i ( i = , 2) (uniform or not) in SU n i , ( R ) are ME then n = n ; the same applies to Sp n i , ( R ) and SO n i , ( R ) . (The higher rank lattices are covered bystronger rigidity statements – see § D ( ) ( G ) = n i ∈ N : 0 < b ( ) i ( G ) < ¥ o SURVEY OF MEASURED GROUP THEORY 17 is a ME-invariant. Conjecture (7) relates this to the dimension of a manifold M in case of G = p ( M ) .One shouldn’t expect dim ( M ) to be an ME-invariant of p ( M ) as the examples of tori show; notealso that for any manifold M one has p ( M × T n ) ME ∼ p ( M × T k ) . However, among negativelycurved manifolds Theorem 3.13 below shows that dim ( M ) is invariant of ℓ -ME.For closed aspherical manifolds M the dimension dim ( M ) is a QI invariant of p ( M ) . Pansuproved that the whole set D ( ) ( G ) is a QI invariant of G . However, positive proportionality of ℓ -Betti numbers for ME fails under QI; in fact, there are QI groups whose Euler characteristics haveopposite signs. Yet Corollary 3.9. For ME groups G and L with well defined Euler characteristic, say fundamentalgroups of compact manifolds, one has c ( L ) = c · c ( G ) , where c = [ G : L ] W ∈ ( , ¥ ) . In particular, the sign (positive, zero, negative) of the Euler characteristic is a ME-invariant. Cowling-Haagerup L -invariant. This numeric invariant L G , taking values in [ , ¥ ] , is defined for any lcsc group G in terms ofnorm bounds on unit approximation in the Fourier algebra A ( G ) (see Cowling and Haagerup [31]).The L -invariant coincides for a lcsc group and its lattices. Moreover, Cowling and Zimmer [32]proved that G OE ∼ G implies L G = L G . In fact, their proof implies the invariance under MeasureEquivalence (see [76]). So L G is a ME-invariant.Cowling and Haagerup [31] computed the L -invariant for simple Lie groups and their lattices: inparticular, proving that L G = G ≃ SO n , ( R ) and SU n , ( R ) , L G = n − G ≃ Sp n , ( R ) , and L G = 21 for the exceptional rank-one group G = F ( − ) .One may observe that simple Lie groups split into two classes: (1) SO n , ( R ) and SU n , ( R ) family,and (2) G ≃ Sp n , ( R ) , F ( − ) and higher rank. Groups in the first class have Haagerup Approxi-mation Property (HAP, a.k.a. a-T-menability) and L G = 1, while groups in the second class haveKazhdan’s property (T) and L G > 1. Cowling conjectured that L G = H ≀ F of a finite group H by the free group F has (HAP), whileOzawa and Popa [108] prove that L H ≀ F > 1. The question whether L G = G is a lattice in G ≃ Sp n , ( R ) or in F ( − ) and L is a lattice ina simple Lie group H , then G ME ∼ L iff G ≃ H . Indeed, higher rank H are ruled out by Zimmer’sTheorem 3.15; H cannot be in the families SO n , ( R ) and SU n , ( R ) by property (T) or Haagerupproperty; and within the family of Sp n , ( R ) and F ( − ) the L -invariant detects G ( ℓ -Betti numberscan also be used for this purpose).3.1.5. Treeability, anti-treeability, ergodic dimension. In [4] Scott Adams introduced the notion of treeable equivalence relations (see § G is Treeable: if there exists an essentially free p.m.p. G -action with a treeable orbit relation. Strongly treeable: if every essentially free p.m.p. G -action gives a treeable orbit relation. Anti-treeable: if there are no essentially free p.m.p. G -actions with a treeable orbit relation.Amenable groups and free groups are strongly treeable. It seems to be still unknown whether thereexist treeable but not strongly treeable groups, in particular it is not clear whether surface groups(which are treeable) are strongly treeable.The properties of being treeable or anti-treeable are ME-invariants. Moreover, G is treeable iff G is amenable (i.e. ME ∼ F = Z ), or is ME to either F or F ¥ (this fact uses Hjorth’s [69], see[81, Theorems 28.2 and 28.5]). Groups with Kazhdan’s property (T) are anti-treeable [6]. Moregenerally, it follows from the recent work of Alvarez and Gaboriau [7] that a non-amenable group G with b ( ) ( G ) = G with C ∗ ( G ) = G -invariant assignment of pointed trees with G as the setof vertices. One may view the relation acting on this measurable family of pointed trees by movingthe marked point. More generally, one might define actions by relations, or measured groupoids, onfields of simplicial complexes. Gaboriau defines (see [57]) the geometric dimension of a relation R to be the smallest possible dimension of such a field of contractible simplicial complexes; the ergodic dimension of a group G will be the minimal geometric dimension over orbit relations R G y X of all essentially free p.m.p. G -actions. In this terminology R is treeable iff it has geometricdimension one, and a group G is treeable if its ergodic dimension is one. There is also a notionof an approximate geometric/ergodic dimension [57] describing the dimensions of a sequence ofsubrelations approximating a given orbit relation. Theorem 3.10 (Gaboriau [57]) . Ergodic dimension and approximate ergodic dimension are ME-invariants. This notion can be used to obtain some information about ME of lattices in the family of rankone groups SO n , ( R ) . If G i < SO n i , ( R ) , i = , G ME ∼ G , then Gaboriau’s resulton ℓ -Betti numbers shows that if one of n i is even, then n = n . However, for n i = k i + b ( ) i vanish. In this case Gaboriau shows, using the above ergodic dimension, that k ≤ k ≤ k or k ≤ k ≤ k .3.1.6. Free products. It was mentioned above that if G i OE ∼ L then ∗ i ∈ I G i OE ∼ ∗ i ∈ I L i (Here G OE ∼ L means that the two groupsadmit an ergodic ME-coupling with index one, equivalently admit essentially free actions which are orbit equivalent ). To what extent does the converse hold? Namely when can one recognize the freefactors on the level of Measure Equivalence?This problem was extensively studied by Ioana, Peterson, and Popa in [75] where strong rigidityresults were obtained for orbit relations under certain assumptions on the actions (see § measurably freely indecomposable groups (MFI) is introduced,and it is shown that this class includes all non-amenable group with b ( ) = 0. Thus infinite property(T) groups, non-amenable direct products, are examples of MFI groups. SURVEY OF MEASURED GROUP THEORY 19 Theorem 3.11 (Alvarez-Gaboriau [7]) . Suppose that ∗ ni = G i ME ∼ ∗ mj = L j , where { G i } ni = and { L j } mj = are two sets of MFI groups with G i ME ∼ G i ′ for ≤ i = i ′ ≤ n, and L j ME ∼ L j ′ for ≤ j = j ′ ≤ m.Then n = m and, up to a permutation of indices, G i ME ∼ L i . Another result from [7] concerning decompositions of equivalence relations as free products ofsub-relations is discussed in § The classes C reg and C . In § second boundedcohomology with unitary coefficients: H b ( G , p ) – a certain vector space associated to a countablegroup G and a unitary representation p : G → U ( H p ) . (Some background on bounded cohomologycan be found in [101, § 3] or [98]; for more details see [21, 97]). Monod and Shalom define the class C reg of groups characterized by the property that H b ( G , ℓ ( G )) = { } and (potentially larger) class C of groups G with non-vanishing H b ( G , p ) for some mixing G -representation p . Known examples of groups in C reg ⊂ C include groups admitting ”hyperbolic-like” actions of thefollowing types: (see [100], [96])(i) non-elementary simplicial action on some simplicial tree, proper on the set of edges;(ii) non-elementary proper isometric action on some proper CAT(-1) space;(iii) non-elementary proper isometric action on some Gromov-hyperbolic graph of bounded va-lency.Hence C reg includes free groups, free products of arbitrary countable groups and free products amal-gamated over a finite group (with the usual exceptions of order two), fundamental groups of neg-atively curved manifolds, Gromov hyperbolic groups, and non-elementary subgroups of the abovefamilies. Examples of groups not in C include amenable groups, products of at least two infinitegroups, lattices in higher rank simple Lie groups (over any local field), irreducible lattices in prod-ucts of general compactly generated non-amenable groups (see [101, § Theorem 3.12 (Monod-Shalom [101]) . (1) Membership in C reg or C is a ME-invariant.(2) For direct products G = G × · · · × G n where G i ∈ C reg are torsion free, the number of factorsand their ME types are ME-invariants.(3) For G as above, if L ME ∼ G then L cannot be written as product of m > n infinite torsion freefactors. Dimension and simplicial volume ( ℓ -ME). Geometric properties are hard to capture with the notion of Measure Equivalence. The ℓ -Bettinumbers is an exception, but this invariant benefits from its Hilbert space nature. In [13, 14] Uri Bader, Roman Sauer and the author consider a restricted version of Measure Equivalence, namely ℓ -ME (see 2.3.2 for definition). Being ℓ -ME is an equivalence relation between finitely generatedgroups, in which any two integrable lattices in the same lcsc group are ℓ -ME. All uniform latticesare integrable, and so are all lattices in SO n , ( R ) ≃ Isom ( H n R ) (see 3.2.4). Theorem 3.13 (Bader-Furman-Sauer [14]) . Let G i = p ( M i ) where M i are closed manifolds whichadmit a Riemannian metric of negative sectional curvature. Assume that G and G admit an ℓ -ME-coupling W . Then dim ( M ) = dim ( M ) and k M k = [ G : G ] W · k M k , where k M i k denotes the simplicial volume of M i . The simplicial volume k M k of a closed manifold M , introduced by Gromov in [65], is the normof the image of the fundamental class under the comparison map H n ( M ) → H ℓ n ( M ) into the ℓ -homology, which is an ℓ -completion of the usual homology. This is a homotopy invariant ofmanifolds. Manifolds carrying a Riemannian metric of negative curvature have k M k > Orbit/Measure Equivalence Rigidity. Let us now turn to Measure Equivalence rigidity results, i.e., classification results in the MEcategory. In the introduction to this section we mentioned that the ME class ME ( Z ) is precisely allinfinite amenable groups. The (distinct) classes ME ( F ≤ n < ¥ ) and ME ( F ¥ ) are very rich and resistprecise description. However, much is known about more rigid families of groups.3.2.1. Higher rank lattices. Theorem 3.14 (Zimmer [139]) . Let G and G ′ be center free simple Lie groups with rk R ( G ) ≥ ,let G < G, G ′ < G ′ be lattices and G y ( X , m ) OE ∼ G ′ y ( X ′ , m ′ ) be Orbit Equivalence betweenessentially free probability measure preserving actions. Then G ∼ = G ′ . Moreover the induced actionsG y ( G × G X ) , G ′ y ( G ′ × G ′ Y ) are isomorphic up to a choice of the isomorphism G ∼ = G ′ . In other words ergodic (infinite) p.m.p. actions of lattices in distinct higher rank semi-simple Liegroups always have distinct orbit structures , for example2 ≤ n < m = ⇒ SL n ( Z ) y T n OE ∼ SL m ( Z ) y T m . This remarkable result (a contemporary of Ornstein-Weiss Theorem 3.1) not only showed that thevariety of orbit structures of non-amenable groups is very rich, but more importantly established alink between OE in Ergodic Theory and the theory of algebraic groups and their lattices; in partic-ular, introducing Margulis’ superrigidity phenomena into Ergodic Theory. This seminal result canbe considered as the birth of the subject discussed in this survey. Let us record a ME conclusion ofthe above. Corollary 3.15 (Zimmer) . Let G, G ′ be connected center free simple Lie groups with rk R ( G ) ≥ , G < G and G ′ < G ′ lattices. Then G ME ∼ G ′ iff G ∼ = G ′ . There is no need here to assume here that the actions are essentially free. Stuck and Zimmer [135] showed that allnon-atomic ergodic p.m.p. actions of higher rank lattices are essentially free; this is based on and generalizes the famousFactor Theorem of Margulis [94], see [95]. SURVEY OF MEASURED GROUP THEORY 21 The picture of ME classes of lattices in higher rank simple Lie groups can be sharpened asfollows. Theorem 3.16 ([44]) . Let G be a center free simple Lie group with rk R ( G ) ≥ , G < G a lattice, L some group Measure Equivalent to G .Then L is commensurable up to finite kernels to a lattice in G. Moreover any ergodic ( G , L ) -coupling has a quotient which is either an atomic coupling (in which case G and L are commensu-rable), or G, or Aut ( G ) with the Haar measure. (Recall that Aut ( G ) contains Ad ( G ) ∼ = G as a finite index subgroup). The main point of thisresult is a construction of a representation r : L → Aut ( G ) for the unknown group L using ME to ahigher rank lattice G . It uses Zimmer’s cocycle superrigidity theorem and a construction involvinga bi- G -equivariant measurable map W × L ˇ W → Aut ( G ) . An updated version of this construction isstated in § F : W → Aut ( G ) satisfying F ( gw ) = g F ( w ) , F ( lw ) = F ( w ) r ( l ) − . It defines the above quotients (the push-forward measure F ∗ m is identified as either atomic, orHaar measure on G ∼ = Ad ( G ) or on all of Aut ( G ) , using Ratner’s theorem [121]). This additionalinformation is useful to derive OE rigidity results (see Theorem 4.19).3.2.2. Products of hyperbolic-like groups. The results above use in an essential way the cocycle superrigidity theorem of Zimmer, whichexploits higher rank phenomena as in Margulis’ superrigidity. A particular situation where suchphenomena take place are irreducible lattices in products of (semi)simple groups, starting fromSL ( R ) × SL ( R ) ; or cocycles over irreducible actions of a product of n ≥ G × · · · × G n y ( X , m ) means ergodicity of G i y ( X , m ) for each 1 ≤ i ≤ n . It recently became clear that higher rank phenomena occur also for irreducible latticesin products of n ≥ n ≥ ”higher rank thrust” tothe situation. The following break-through results of Nicolas Monod and Yehuda Shalom is anexcellent illustration of this fact (see § Theorem 3.17 (Monod-Shalom [101, Theorem 1.16]) . Let G = G × · · · × G n and L = L × · · · × L m be products of torsion-free countable groups, where G i ∈ C reg . Assume that G ME ∼ L .Then n ≥ m. If n = m then, after a permutation of the indices, G i ME ∼ L i . In the latter case (n = m)any ergodic ME-coupling of G ∼ = L has the trivial coupling as a quotient. Theorem 3.18 (Monod-Shalom [101]) . Let G = G × · · · × G n where n ≥ and G i are torsion freegroups in class C , and G y ( X , m ) be an irreducible action (i.e., every G i y ( X , m ) is ergodic); let L be a torsion free countable group and L y ( Y , n ) be a mildly mixing action. If G y X SOE ∼ L y Y ,then this SOE has index one, L ∼ = G and the actions are isomorphic. Sometimes this can be relaxed to ergodicity of G ′ i y ( X , m ) where G ′ i = (cid:213) j = i G j . Theorem 3.19 (Monod-Shalom [101]) . For i = , let → A i → ¯ G i → G i → be short exactsequence of groups with A i amenable and G i are in C reg and are torsion free. Then ¯ G ME ∼ ¯ G implies G ME ∼ G . A key tool in the proofs of these results is a cocycle superrigidity theorem 5.5, which involves sec-ond bounded cohomology H b of groups. In [12] (see also [15]) Uri Bader and the author develop adifferent approach to higher rank phenomena, in particular showing an analogue of Monod-ShalomTheorem 5.5, as stated in Theorem 5.6. This result concerns a class of groups which admit con-vergence action on a compact metrizable space (i.e. a continuous action H y M where the action H y M \ Diag on the locally compact space of distinct triples is proper). Following Furstenberg[50] we denote this class as D , and distinguish a subclass D ea of groups admitting convergent action H y M with amenable stabilizers. As a consequence of this superrigidity theorem it follows thatTheorems 3.17–3.19 remain valid if class C reg is replaces by D ea . Recently Hiroki Sako [122, 123]has obtained similar results for groups in Ozawa’s class S (see [107]).Let us point out that each of the classes C reg , D ea , S include all Gromov hyperbolic groups (andmany relatively hyperbolic ones), are closed under taking subgroups, and exclude direct products oftwo infinite groups. These are key features of what one would like to call ”hyperboli-like” group.3.2.3. Mapping Class Groups. The following remarkable result of Yoshikata Kida concerns Mapping Class Groups of surfaces.Given a compact orientable surface S g , p of genus g with p boundary components the extendedmapping class group G ( S g , p ) ⋄ is the group of isotopy components of diffeomorphisms of S g , p (the mapping class group itself is the index two subgroup of isotopy classes of orientation preservingdiffeomorphisms). In the following assume 3 g + p > 0, i.e., rule out the torus S , , once puncturedtorus S , , and spheres S , p with p ≤ Theorem 3.20 (Kida [86]) . Let G be a finite index subgroup in G ( S g , p ) ⋄ with g + p − > , or ina finite product of such Mapping Class groups (cid:213) ni = G ( S g , p ) ⋄ .Then any group L ME ∼ G is commensurable up to finite kernels to G , and ergodic ME-coupling hasa discrete ( G , L ) -coupling as a quotient. This work (spanning [83, 85, 86]) is a real tour de force. Mapping Class Groups G ( S ) are oftencompared to a lattice in a semi-simple Lie group G : the Teichm¨uller space T ( S ) is analogous to thesymmetric space G / K , Thurston boundary PML ( S ) analogous to Furstenberg boundary B ( G ) = G / P , and the curve complex C ( S ) to the spherical Tits building of G . The MCG has been extensivelystudied as a geometric object, while Kida’s work provides a new ergodic-theoretic perspective. Forexample, Kida proves that Thurston boundary PML ( S ) with the Lebesgue measure class is G - boundary in the sense of Burger-Monod for the Mapping Class Group, i.e., the action of the latteris amenable and doubly ergodic with unitary coefficients. Properties of the MCG action on PML ( S ) allow Kida to characterize certain subrelations/subgroupoids arising in self Measure Equivalence ofa MCG; leading to the proof of a cocycle (strong) rigidity theorem 5.7, which can be viewed as agroupoid version of Ivanov’s rigidity theorem. This strong rigidity theorem can be used with § SURVEY OF MEASURED GROUP THEORY 23 of MCGs without any irreducibility assumptions. From this point of view MCGs are more MErigid than higher rank lattices, despite the fact that they lack many other rigidity attributes, such asproperty (T) (see Andersen [10]). Added in proof . Very recently additional extremely strong ME-rigidity results were obtained inKida [88] and Popa and Vaes [120] for certain amalgamated products of higher rank lattices, andthe also mapping class groups. The latter paper also establishes W ∗ -rigidity.3.2.4. Hyperbolic lattices and ℓ -ME. Measure Equivalence is motivated by the theory of lattices, with ME-couplings generalizing thesituation of groups imbedded as lattices in in the same ambient lcsc group. Thus, in the contextof semi-simple groups, one wonders whether ME rigidity results would parallel Mostow rigid-ity; and in particular would apply to (lattices in) all simple groups with the usual exception ofSL ( R ) ≃ SO , ( R ) ≃ SU , ( R ) . The higher rank situation (at least that of simple groups) is wellunderstood ( § n , ( R ) , SU m , ( R ) , Sp k , ( R ) ,and F ( − ) ) known ME-invariants discussed above (namely: property (T), ℓ -Betti numbers, L -invariant, ergodic dimension) allow to distinguish lattices among most rank one groups. This refersto statements of the form: if G i < G i are lattices then G ME ∼ G iff G ≃ G . However ME classifica-tion such as in Theorems 3.16, 3.17, 3.20 are not known for rank one cases. The ingredient whichis missing in the existing approach is an appropriate cocycle superrigidity theorem .In a joint work with Uri Bader and Roman Sauer a cocycle strong rigidity theorem is provedfor ME-cocycles for lattices in SO n , ( R ) ≃ Isom ( H n R ) , n ≥ 3, under a certain ℓ -assumption (see § Theorem 3.21 (Bader-Furman-Sauer [13]) . Let G is a lattice in G = Isom ( H n ) , n ≥ , and L issome finitely generated group ℓ -ME to G then L is a lattice in G modulo a finite normal subgroup.Moreover any ergodic ( G , L ) -coupling has a quotient, which is ether discrete, or G = Aut ( G ) , orG with the Haar measure. Recently Sorin Popa has introduced a new set of ideas for studying Orbit Equivalence. Theseresults, rather than relying on rigidity of the acting groups alone, exploit rigidity aspects of groupsactions of certain type. We shall discuss them in §§ How many Orbit Structures does a given group have? Theorem 3.1 of Ornstein and Weiss [105] implies that for an infinite amenable countable group G all ergodic probability measure preserving actions G y ( X , m ) define the same orbit structure,namely R amen . What happens for non-amenable groups G ? Theorem 3.22 (Epstein [36], after Ioana [73] and Gabotiau-Lyons [60]) . Any non-amenable count-able group G has a continuum of essentially free ergodic probability measure preserving actions G y ( X , m ) , no two of which are stably Orbit Equivalent. For Sp n , ( R ) and F ( − ) a cocycle superrigidity theorem was proved by Corlette and Zimmer [29] (see also Fisherand Hitchman [43]), but these results requires boundness assumptions which preclude them from being used for ME-cocycles. Let us briefly discuss the problem and its solution. Since Card ( Aut ( X , m ) G ) = (cid:192) = (cid:192) thereare at most continuum many actions for a fixed countable group G . The fact, this upper bound onthe cardinality of isomorphism classes of actions is achieved, using the corresponding fact aboutunitary representations and the Gaussian construction. Hence one might expect at most (cid:192) -manynon-OE actions for any given G . OE rigidity results showed that some specific classes of groupsindeed have this many mutually non-OE actions; this includes: higher rank lattices [64], productsof hyperbolic-like groups [101, Theorem 1.7], and some other classes of groups [110, 113]). But thegeneral question, regarding an arbitrary non-amenable G , remained open.Most invariants of equivalence relations depend on the acting group rather than the action, andthus could not be used to distinguish actions of the fixed group G . The notable exception to thismeta-mathematical statement appears for non-amenable groups which do not have property (T).For such groups two non-SOE actions can easily be constructed: (1) a strongly ergodic action(using Schmidt’s [129]) and (2) an ergodic action which is not strongly ergodic (using Connes-Weiss [28]). Taking a product with an essentially free weakly mixing strongly ergodic G -actions(e.g. the Bernoulli action ( X , m ) G ) one makes the above two actions essentially free and distinct.In [68] Greg Hjorth showed that if G has property (T) the set of isomorphism classes of orbitstructures for essentially free G -actions has cardinality (cid:192) , by proving that the natural map fromthe isomorphism classes of essentially free ergodic G -actions to the isomorphism classes of G -orbitstructures is at most countable-to-one. More precisely, the space of G -actions producing a fixed orbitstructure is equipped with a structure of a Polish space (separability) where any two nearby actionsare shown to be conjugate. This is an example of proving rigidity up to countable classes combining separability of the ambient space with a local rigidity phenomenon (stemming from property (T),see § § (cid:192) -many, non-OE essentially free ergodic actions. Damien Gaboriau and Sorin Popa [59] achieved thisgoal for the quintessential representative of a non-amenable group without property (T), namely forthe free group F . Using a sophisticated rigidity vs. separability argument they showed that withina certain rich family of F -actions the map from isomorphism classes of actions to orbit structures iscountable-to-one. The rigidity component of the argument was this time provided by Popa’s notionof w -rigid actions such as SL ( Z ) y T , with the rigidity related to the relative property (T) for thesemi-direct product SL ( Z ) ⋉ Z viewing Z as the Pontryagin dual of T .In [73] Adrian Ioana obtained a sweeping result showing that any G containing a copy of F has (cid:192) -many mutually non-SOE essentially free actions. The basic idea of the construction being touse a family of non-SOE of F -actions F y X t to construct co-induced G -actions G y X G / F t andpushing the solution of F -problem to the analysis of the co-induced actions. The class of groupscontaining F covers ”most of” the class of non-amenable groups with few, very hard to obtain,exceptions. The ultimate solution to the problem, covering all non-amenable groups, was shortlyobtained by Inessa Epstein [36] using a result by Damien Gaboriau and Russel Lyons [60], whoproved that any non-amenable G contains a F in a sort of measure-theoretical sense. Epstein wasable to show that this sort of containment suffices to carry out an analogue of Ioana’s co-inductionargument [73] to prove Theorem 3.22. SURVEY OF MEASURED GROUP THEORY 25 Furthermore, in [74] Ioana, Kechris, Tsankov jointly with Epstein show that for any non-amenable G the space of all ergodic free p.m.p. actions taken up to OE not only has cardinality of the con-tinuum, but is also impossible to classify in a very strong sense. One may also add, that most ofthe general results mentioned above show that within certain families of actions the grouping intoSOE-ones has countable classes, therefore giving only implicit families of non-SOE actions. In [71]Ioana provided an explicit list of a continuum of mutually non-SOE actions of F . 4. M EASURED EQUIVALENCE RELATIONS Basic definitions. We start with the notion of countable equivalence relations in the Borel setting. It consists ofa standard Borel space ( X , X ) (cf. [39] for definitions) and a Borel subset R ⊂ X × X which is anequivalence relation, whose equivalence classes R [ x ] = { y ∈ X : ( x , y ) ∈ R } are all countable.To construct such relations choose a countable collection F = { f i } i ∈ I of Borel bijections f i : A i → B i between Borel subsets A i , B i ∈ X , i ∈ I ; and let R F be the smallest equivalence relationincluding the graphs of all f i , i ∈ I . More precisely, ( x , y ) ∈ R F iff there exists a finite sequence i , . . . , i k ∈ I and e , . . . , e k ∈ {− , } so that y = f e k i k ◦ · · · ◦ f e i ◦ f e i ( x ) . We shall say that the family F generates the relation R F . The particular case of a collection F = { f i } of Borel isomorphisms of the whole space X generates a countable group G = h F i and R F = R G y X = { ( x , y ) : G x = G y } = { ( x , g . x ) : x ∈ X , g ∈ G } . Feldman and Moore [39] proved that any countable Borel equivalence relation admits a generatingset whose elements are defined on all of X ; in other words, any equivalence relation appears as theorbit relation R G y X of a Borel action G y X of some countable group G (see 4.3.1).Given a countable Borel equivalence relation R the full group [ R ] is defined by [ R ] = { f ∈ Aut ( X , X ) : ∀ x ∈ X : ( x , f ( x )) ∈ R } . The full pseudo-group [[ R ]] consists of partially defined Borel isomorphisms y : Dom ( y ) → Im ( y ) , so that Graph ( y ) = { ( x , y ( x )) : x ∈ Dom ( y ) } ⊂ R . If R is the orbit relation R G y X of a group action G y ( X , X ) , then any f ∈ [ R ] has the following”piece-wise G -structure”: there exist countable partitions F A i = X = F B i into Borel sets and ele-ments g i ∈ G with g i ( A i ) = B i so that f ( x ) = g i x for x ∈ A i . Elements y of the full pseudo-group [[ R G ]] have a similar ”piece-wise G -structure” with F A i = Dom ( y ) and F B i = Im ( y ) .Let R be a countable Borel equivalence relation on a standard Borel space ( X , X ) . A measure m on ( X , X ) is R - invariant (resp. R - quasi-invariant ) if for all f ∈ [ R ] , f ∗ m = m (resp. f ∗ m ∼ m ).Note that if F = { f i : A i → B i } is a generating set for R then m is R -invariant iff m is invariant undereach f i , i.e. m ( f − i ( E ) ∩ A i ) = m ( E ∩ B i ) for all E ∈ X . Similarly, quasi-invariance of a measure canbe tested on a generating set. The R - saturation of E ∈ X is R [ E ] = { x ∈ X : ∃ y ∈ E , ( x , y ) ∈ R } .A R (quasi-) invariant measure m is ergodic if R [ E ] is either m -null or m -conull for any E ∈ X . Inthis section we shall focus on countable Borel equivalence relations R on ( X , X ) equipped with anergodic, invariant, non-atomic, probability measure m on ( X , X ) . Such a quadruple ( X , X , m , R ) iscalled type II -relation. These are precisely the orbit relations of ergodic measure preserving actionsof countable groups on non-atomic standard probability measure spaces (the non-trivial implicationfollows from the above mentioned theorem of Feldman and Moore). SURVEY OF MEASURED GROUP THEORY 27 Given a countable Borel relation R on ( X , X ) and an R -quasi-invariant probability measure m ,define infinite measures ˜ m L , ˜ m R on R by˜ m L ( E ) = Z X { y : ( x , y ) ∈ E ∩ R } d m ( x ) , ˜ m R ( E ) = Z X { x : ( x , y ) ∈ E ∩ R } d m ( y ) . These measures are equivalent, and coincide if m is R -invariant, which is our main focus. In thiscase we shall denote(4.1) ˜ m = ˜ m L = ˜ m R Hereafter, saying that some property holds a.e. on R would refer to ˜ m -a.e. (this makes sense evenif m is only R -quasi-invariant). Remark 4.1. In some situations a Borel Equivalence relation R on ( X , X ) has only one (non-atomic) invariant probability measure. For example, this is the case for the orbit relation of thestandard action of a finite index subgroup G < SL n ( Z ) on the torus T n = R n / Z n , or for a lattice G in a simple center free Lie group G acting on H / L , where H is a simple Lie group, L < H is a lattice,and G acts by left translations via an embedding j : G → H with j ( G ) having trivial centralizer in H . In such situations one may gain understanding of the countable Borel equivalence relation R via the study of the II -relation corresponding to the unique R -invariant probability measure.As always in the measure-theoretic setting null sets should be considered negligible. So anisomorphism T between (complete) measure spaces ( X i , X i , m i ) , i = , 2, is a Borel isomorphismbetween m i -conull sets T : X ′ → X ′ with T ∗ ( m ) = m . In the context of II -relations, we declaretwo relations ( X i , X i , m i , R i ) , i = , isomorphic , if the exists a measure space isomorphism T : ( X , m ) ∼ = ( X , m ) so that T × T : ( R , ˜ m ) → ( R , ˜ m ) is an isomorphism. In other words, aftera restriction to conull sets, T satisfies ( x , y ) ∈ R ⇐⇒ ( T ( x ) , T ( y )) ∈ R . Let us also adapt the notions of the full group and the full pseudo-group to the measure-theoreticsetting, by passing to a quotient Aut ( X , X ) → Aut ( X , X , m ) where two Borel isomorphism f and f ′ which agree m -a.e. are identified. This allows us to focus on the essential measure-theoretic issues.The following easy, but useful Lemma illustrates the advantage of this framework. Lemma 4.2. Let ( X , X , m , R ) be a II -relation. Then for A , B ∈ X one has m ( f ( A ) △ B ) = forsome f ∈ [ R ] iff m ( A ) = m ( B ) . Restriction and weak isomorphisms. Equivalence relations admit a natural operation of restriction , sometimes called induction , to asubset: given a relation R on X and a measurable subset A ⊂ X the restriction R A to A is R A = R ∩ ( A × A ) . Or just Zariski dense subgroup, see [18]. In the presence of, say R -invariant, measure m on ( X , X ) the restriction to a subset A ⊂ X with m ( A ) > m | A , defined by m | A ( E ) = m ( A ∩ E ) . If m is a probabilitymeasure, we shall denote by m A the normalized restriction m A = m ( A ) − · m | A . It is easy to see thatergodicity is preserved, so a restriction of a II -relation ( X , m , R ) to a positive measure subset A ⊂ X is a II -relation ( A , m A , R A ) . Remark 4.3. Note that it follows from Lemma 4.2 that the isomorphism class of R A dependsonly on R and on the size m ( A ) , so R A may be denoted R t where t = m ( A ) is 0 < t ≤ 1. Onemay also define R t for t > 1. For an integer k > R k denote the product of R with the fullrelation on the finite set { , . . . , k } , namely the relation on X × { , . . . , k } with (( x , i ) , ( y , j )) ∈ R k iff ( x , y ) ∈ R . So ( R k ) / k ∼ = R ∼ = R . The definition of R t can now be extended to all 0 < t < ¥ using an easily verified formula ( R t ) s ∼ = R ts . This construction is closely related to the notionof an amplification in von-Neumann algebras: the Murray von Neumann group-measure spaceconstruction M R satisfies M R t = ( M R ) t .The operation of restriction/induction allows one to relax the notion of isomorphism of II -relations as follows: Definition 4.4. Two II -relations R and R are weakly isomorphic if R ∼ = R t for some t ∈ R × + .Equivalently, if there exist positive measurable subsets A i ⊂ X i with m ( A ) = t · m ( A ) and anisomorphism between the restrictions of R i to A i .Observe that two ergodic probability measure-preserving actions G i y ( X i , X i , m i ) of countablegroups are orbit equivalent iff the corresponding orbit relations R G i y X i are isomorphic.4.2. Invariants of equivalence relations. Let us now discuss in some detail several qualitative and numerical properties of II equivalencerelations which are preserved under isomorphisms and often preserved or rescaled by the indexunder weak isomorphisms. We refer to such properties as invariants of equivalence relations. Manyof these properties are motivated by properties of groups, and often an orbit relation R G y X of anessentially free action of countable group would be a reflection of the corresponding property of G .4.2.1. Amenability, strong ergodicity, property (T). Amenability of an equivalence relation can be defined in a number of ways. In [138] Zimmerintroduced the notion of amenability for a group action on a space with quasi-invariant measure.This notion plays a central role in the theory. This definition is parallel to the fixed point charac-terization of amenability for groups. For equivalence relation R on ( X , X ) with a quasi-invariant measure m it reads as follows.Let E be a separable Banach space, and c : R → Isom ( E ) be a measurable 1- cocycle , i.e., ameasurable (with respect to the weak topology on E ) map, satisfying ˜ m -a.e. c ( x , z ) = c ( x , y ) ◦ c ( y , z ) . Let X ∋ x Q x ⊂ E ∗ be a measurable family of non-empty convex compact subsets of the dual space E ∗ taken with the ∗ -topology, so that c ( x , y ) ∗ ( Q x ) = Q y . The relation R is amenable if any such SURVEY OF MEASURED GROUP THEORY 29 family contains a measurable invariant section, i.e., a measurable assignment X ∋ x p ( x ) ∈ Q x ,so that a.e. c ( x , y ) ∗ p ( x ) = p ( y ) . The (original) definition of amenability for group actions concerned general cocycles c : G × X → Isom ( E ) rather than the ones depending only on the orbit relation R G y X . The language of measuredgroupoids provides a common framework for both settings (see [9]).Any non-singular action of an amenable group is amenable, because any cocycle c : G × X → Isom ( E ) can be used to define an affine G -action on the closed convex subset of L ¥ ( X , E ∗ ) = L ( X , E ) ∗ consisting of all measurable sections x → p ( x ) ∈ Q x ; the fixed point property of G pro-vides the desired c ∗ -invariant section. The converse is not true: any (countable, or lcsc) groupadmits essentially free amenable action with a quasi-invariant measure – this is the main use of thenotion of amenable actions. However, for essentially free, probability measure preserving actions,amenability of the II -relation R G y X implies (hence is equivalent to) amenability of G . Indeed,given an affine G action a on a convex compact Q ⊂ E ∗ , one can take Q x = Q for all x ∈ X and set c ( gx , x ) = a ( g ) ; amenability of R G y X provides an invariant section p : X → Q whose barycenter q = R X p ( x ) d m ( x ) would be an a ( G ) -fixed point in Q .Connes, Feldman and Weiss [27] proved that amenable relations are hyperfinite in the sensethat they can be viewed as an increasing union of finite subrelations; they also showed that sucha relation can be generated by an action of Z (see also [78] by Kaimanovich for a nice expositionand several other nice characterizations of amenability). It follows that there is only one amenableII -relation, which we denote hereafter by R amen . In [141] Zimmer introduced the notion of property (T) for group actions on measure spaces withquasi-invariant measure. The equivalence relation version can be stated as follows. Let H be aseparable Hilbert space and let c : R → U ( H ) be a measurable 1- cocycle , i.e., c satisfies c ( x , z ) = c ( x , y ) ◦ c ( y , z ) Then R has property (T) if any such cocycle for which there exists a sequence v n : X → S ( H ) ofmeasurable maps into the unit sphere S ( H ) with k v n ( y ) − c ( x , y ) v n k → [ ˜ m ] -a.e.admits a measurable map u : X → S ( H ) with u ( y ) = c ( x , y ) u ( x ) for ˜ m -a.e. ( x , y ) ∈ R . For anessentially free probability measure preserving action G y ( X , m ) the orbit relation R G y X has prop-erty (T) if and only if the group G has Kazhdan’s property (T) (in [141] weak mixing of the actionwas assumed for the ”only if” implication, but this can be removed as in § R be a II -equivalence relation on ( X , m ) . A sequence { A n } of measurable subsets of X is asymptotically R -invariant , if m ( f ( A n ) △ A n ) → f ∈ [ R ] . This is satisfied triviallyif m ( A n ) · ( − m ( A n )) → 0. Relation R is strongly ergodic if any asymptotically R -invariantsequence of sets is trivial in the above sense. (Note that the condition of asymptotic invariance maybe checked on elements f i of any generating system F of R ). The amenable relation R amen is not strongly ergodic. If an action G y ( X , m ) has a spectral gap (i.e., does not have almost invariant vectors) in the Koopman representation on L ( X , m ) ⊖ C then R G y X is strongly ergodic. Using the fact that the Koopman representation of a Bernoulli action G y ( X , m ) G is contained in a multiple of the regular representation ¥ · ℓ ( G ) , Schmidt [128]characterized non-amenable groups by the property that they admit p.m.p. actions with stronglyergodic orbit relation. If R is not strongly ergodic then it has an amenable relation as a non-singularquotient (Jones and Schmidt [77]). Connes and Weiss [28] showed that all p.m.p. actions of a group G have strongly ergodic orbit relations if and only if G has Kazhdan’s property (T). In this shortelegant paper they introduced the idea of Gaussian actions as a way of constructing a p.m.p. action from a given unitary representation .In general strong ergodicity of the orbit relation R G y X does not imply a spectral gap for theaction G y ( X , m ) ([128], [70]). However, this implication does hold for generalized Bernoulliactions (Kechris and Tsankov [82]), and when the action has an ergodic centralizer (Chifan andIoana [25, Lemma 10]).4.2.2. Fundamental group - index values of self similarity. The term fundamental group of a II -relation R refers to a subgroup of R × + defined by F ( R ) = (cid:8) t ∈ R × + : R ∼ = R t (cid:9) . Equivalently, for R on ( X , m ) , the fundamental group F ( R ) consists of all ratios m ( A ) / m ( B ) where A , B ⊂ X are positive measure subsets with R A ∼ = R B (here one can take one of the setsto be X without loss of generality). The notion is borrowed from similarly defined concept of thefundamental group of a von Neumann algebra, introduced by Murray and von Neumann [104]: F ( M ) = (cid:8) t ∈ R × + : M t ∼ = M (cid:9) . However, the connection is not direct: even for group space con-struction M = G ⋉ L ¥ ( X ) isomorphisms M ∼ = M t (or even automorphisms of M ) need not respectthe Cartan subalgebra L ¥ ( X ) in general.Since the restriction of the amenable relation R amen to any positive measure subset A ⊂ X isamenable, it follows F ( R amen ) = R × + . The same obviously applies to the product of any relation with an amenable one.On another extreme are orbit relations R G y X of essentially free ergodic action of ICC groups G with property (T): for such relations the fundamental group F ( R G y X ) is at most countable (Gefterand Golodets [64, Corollary 1.8]).Many relations have trivial fundamental group. This includes all II relations with a non-trivialnumeric invariant which scales under restriction:(1) Relations with 1 < cost ( R ) < ¥ ; in particular, orbit relation R G y X for essentially freeactions of F n , 1 < n < ¥ , or surface groups.(2) Relations with some non-trivial ℓ -Betti number 0 < b ( ) i ( R ) < ¥ for some i ∈ N ; in par-ticular, orbit relation R G y X for essentially free actions of a group G with 0 < b ( ) i ( G ) < ¥ for some i ∈ N , such as lattices in SO n , ( R ) , SU m , ( R ) , Sp k , ( R ) .Triviality of the fundamental group often appears as a by-product of rigidity of groups and groupactions. For examples F ( R G y X ) = { } in the following situations: SURVEY OF MEASURED GROUP THEORY 31 (1) Any (essentially free) action of a lattice G in a simple Lie group of higher rank ([64]);(2) Any essentially free action of (finite index subgroups of products of) Mapping Class Groups([86]);(3) Actions of G = G × · · · × G n , n ≥ 2, of hyperbolic-like groups G i where each of them actsergodically ([101]);(4) G dsc -cocycle superrigid actions G y X such as Bernoulli actions of groups with property(T) ([111, 112, 117]).What are other possibilities for the fundamental group beyond the two extreme cases F ( R ) = R × + and F ( R ) = { } ? The most comprehensive answer (to date) to this question is contained inthe following result of S.Popa and S.Vaes (see [118] for further references): Theorem 4.5 (Popa-Vaes, [118, Thm 1.1]) . There exists a family S of additive subgroups of R whichcontains all countable groups, and (uncountable) groups of arbitrary Hausdorff dimension in ( , ) ,so that for any F ∈ S and any totally disconnected locally compact unimodular group G there existuncountably many mutually non-SOE essentially free p.m.p. actions of F ¥ whose orbit relations R = R F ¥ y X have F ( R ) ∼ = exp ( F ) and Out ( R ) ∼ = G.Moreover, in these examples the Murray von Neumann group space factor M = G ⋊ L ¥ ( X ) has F ( M ) ∼ = F ( R ) ∼ = exp ( F ) and Out ( M ) ∼ = Out ( R ) ⋉ H ( R , T ) , where H ( R , T ) is the first coho-mology with coefficients in the -torus. Treeability. An equivalence relation R is said treeable (Adams [4]) if it admits a generating set F = { f i } sothat the corresponding (non-oriented) graph on a.e. R -class is a tree. Basic examples of treeablerelations include: R amen viewing the amenable II -relation as the orbit relation of some/any actionof Z = F , and more generally, R F n y X where F n y X is an essentially free action of the free group F n , 1 ≤ n ≤ ¥ . Any restriction of a treeable relation is treeable, and R is treeable iff R t is.If R → R is a (weak) injective relation morphism and R is treeable, then so is R – the ideais to lift a treeing graphing from R to R piece by piece. This way, one shows that if a group L admits an essentially free action L y Z with treeable R L y Z , and G and L admit (S)OE essentiallyfree actions G y X and L y Y then the G -action on X × Z , g : ( x , z ) ( gx , a ( g , x ) z ) via the (S)OEcocycle a : G × X → L , has a treeable orbit structure R G y X × Z . Since surface groups G = p ( S g ) , g ≥ 2, and F are lattices in PSL ( R ) , hence ME, the former groups have free actions with treeableorbit relations. Are all orbit relations of free actions of a surface group treeable?4.2.4. Cost. The notion of cost for II -relations corresponds to the notion of rank for discretecountable groups. The notion of cost was introduced by G. Levitt [90] and extensively studied byD. Gaboriau [53, 54, 60]. Definition 4.6. Given a generating system F = { f i : A i → B i } i ∈ N for a II -equivalence relation R on ( X , m ) the cost of the graphing F iscost ( F ) = (cid:229) i m ( A i ) = (cid:229) i m ( B i ) and the cost of the relation iscost ( R ) = inf { cost ( F ) : F generates R } . A generating system F defines a graph structure on every R -class and cost ( F ) is half of theaverage valency of this graph over the space ( X , m ) .The cost of a II -relation takes values in [ , ¥ ] . In the definition of the cost of a relation it isimportant that the relation is probability measure preserving, but ergodicity is not essential. Thebroader context includes relations with finite classes, such relations can values less than one. Forinstance, the orbit relation of a (non-ergodic) probability measure preserving action of a finite group G y ( X , m ) one gets cost ( R G y X ) = − | G | . If R is the orbit relation of some (not necessarily free) action G y ( X , m ) then cost ( R ) ≤ rank ( G ) ,where the latter stands for the minimal number of generators for G . Indeed, any generating set { g , . . . , g k } for G gives a generating system F = { g i : X → X } ki = for R G y X . Recall that theamenable II -relation R amen can be generated by (any) action of Z . Hencecost ( R amen ) = . The cost behaves nicely with respect to restriction: Theorem 4.7 (Gaboriau [54]) . For a II -relation R :t · ( cost ( R t ) − ) = cost ( R ) − ( t ∈ R × + ) . The following is a key tool for computations of the cost: Theorem 4.8 (Gaboriau [54]) . Let R be a treeable equivalence relation, and F be a graphing of R giving a tree structure to R -classes. Then cost ( R ) = cost ( F ) . Conversely, for a relation R with cost ( R ) < ¥ , if the cost is attained by some graphing Y then Y is a treeing of R . The above result (the first part) implies that for any essentially free action F n y ( X , m ) one hascost ( R F n y X ) = n . This allowed Gaboriau to prove the following fact, answering a long standingquestion: Corollary 4.9 (Gaboriau [53, 54]) . If essentially free probability measure preserving actions of F n and F m are Orbit Equivalent then n = m. Note that F n and F m are commensurable for 2 ≤ n , m < ¥ , hence they have essentially freeactions which are weakly isomorphic . The index of such weak isomorphism will necessarily be n − m − , or m − n − (these free groups have P.F. - fixed price). It should be pointed out that one of the majoropen problems in the theory of von Neumann algebras is whether it is possible for the factors L ( F n ) and L ( F m ) to be isomorphic for n = m (it is known that either all L ( F n ) , 2 ≤ n < ¥ , are isomorphic,or all distinct).The following powerful result of Greg Hjorth provides a link from treeable relations back toactions of free groups: Theorem 4.10 (Hjorth [69]) . Let R be a treeable equivalence relation with n = cost ( R ) in { , , . . . , ¥ } .Then R can be generated by an essentially free action of F n . SURVEY OF MEASURED GROUP THEORY 33 The point of the proof is to show that a relation R which has a treeing graphing with averagevalency n admits a (treeing) graphing with a.e. constant valency 2 n .The behavior of the cost under passing to a subrelation of finite index is quite subtle – the follow-ing question is still open (to the best of author’s knowledge). Question 4.11 (Gaboriau) . Let G ′ be a subgroup of finite index in G , and G y ( X , m ) be an essen-tially free p.m.p. action. Is it true that the costs of the orbit relations of G and G ′ are related by theindex [ G : G ′ ] : cost ( R G ′ y X ) − = [ G : G ′ ] · ( cost ( R G y X ) − ) ?In general G ′ has at most [ G : G ′ ] -many ergodic components. The extreme case where the numberof G ′ -ergodic components is maximal: [ G : G ′ ] , corresponds to G y ( X , m ) being co-induction froman ergodic G ′ -action. In this case the above formula easily holds. The real question lies in the otherextreme where G ′ is ergodic.Recall that the notion of the cost is analogous to the notion of rank for groups, where rank ( G ) = inf { n ∈ N : ∃ epimorphism F n → G } . Schreier’s theorem states that for n ∈ N any subgroup F < F n of finite index [ F n : F ] = k is itself free: F ∼ = F k ( n − )+ . This implies that for any finitely generated G and any finite index subgroup of G ′ < G one hasrank ( G ′ ) − ≤ [ G : G ′ ] · ( rank ( G ) − ) with equality in the case of free groups. Let G > G > . . . be a chain of subgroups of finite index.One defines the rank gradient (Lackenby [89]) of the chain { G n } as the limit of the monotonic (!)sequence: RG ( G , { G n } ) = lim n → ¥ rank ( G n ) − [ G : G n ] . It is an intriguing question whether (or when) is it true that RG ( G , { G n } ) depends only on G and noton a particular chain of finite index subgroups. One should, of course, assume that the chains inquestion have trivial intersection, and one might require the chains to consists of normal subgroupsin the original group. In the case of free groups RG is indeed independent of the chain.In [1] Abert and Nikolov prove that the rank gradient of a chain of finite index subgroups of G is given by the cost of a certain associated ergodic p.m.p. G -action. Let us formulate a specialcase of this relation where the chain { G n } consists of normal subgroups G n with T G n = { } . Let K = lim ←− G / G n denote the profinite completion corresponding to the chain. The G -action by lefttranslations on the compact totally disconnected group K preserves the Haar measure m K and G y ( K , m K ) is a free ergodic p.m.p. action. (let us point out in passing that this action has a spectralgap, implying strong ergodicity, iff the chain has property ( t ) introduced by Lubotzky and Zimmer[91]). Theorem 4.12 (Abert-Nikolov [1]) . With the above notations: RG ( G , { G n } ) = cost ( R G y K ) − . One direction ( ≥ ) is easy to explain. Let K n be the closure of G n in K . Then K n is an open normalsubgroup of K of index m = [ G : G n ] . Let 1 = g , g , . . . , g n ∈ G be representatives of G n -cosets, and h , . . . , h k generators of G n with k = rank ( G n ) . Consider the graphing F = { f , . . . , f m , y , . . . , y k } , where f i : K n → g i K n are restrictions of g i (2 ≤ i ≤ m ), and y j : K n → K n are restrictions of h j (1 ≤ j ≤ k ). These maps are easily seen to generate R G y K , with the cost ofcost ( F ) = k · m K ( K n ) + ( m − ) · m K ( K n ) = k − m + = + rank ( G n ) − [ G : G n ] . Abert and Nikolov observed that a positive answer to Question 4.11 combined with the aboveresult shows that RG ( G ) is independent of the choice of a (normal) chain, and therefore is a numericinvariant associated to any residually finite finitely generated groups. Surprisingly, this turns outto be related to a problem in the theory of compact hyperbolic 3-manifolds concerning rank versusHeegard genus [89] – see [1] for the connection and further discussions.The above result has an application, independent of Question 4.11. Since amenable groups haveP.F. with C = 1, it follows that a finitely generated, residually finite amenable group G has sub-linearrank growth for finite index normal subgroups with trivial intersection, i.e., RG ( G ) = ℓ -Betti numbers. We have already mentioned the ℓ -Betti numbers b ( ) i ( G ) associated with a discrete group G andGaboriau’s proportionality result 3.8 for Measure Equivalence between groups. In fact, rather thanrelating the ℓ -Betti numbers of groups via ME, in [57] Gaboriau- defines the notion of ℓ -Betti numbers b ( ) i ( R ) for a II -equivalence relation R ;- proves that b ( ) i ( G ) = b ( ) i ( R G y X ) for essentially free ergodic action G y ( X , m ) ;- observes that b ( ) i ( R t ) = t · b ( ) i ( R ) for any II -relation.The definition of b ( ) i ( R ) is inspired by the definition of b ( ) i ( G ) by Cheeger and Gromov [23]:it uses R -action (or groupoid action) on pointed contractible simplicial complexes, correspondingcomplexes of Hilbert modules with R -action, and von-Neumann dimension with respect to thealgebra M R .In the late 1990s Wolgang L ¨uck developed an algebraic notion of dimension for arbitrary mod-ules over ven Neumann algebras, in particular giving an alternative approach to ℓ -Betti numbersfor groups (see [92]). In [124] Roman Sauer used L ¨uck’s notion of dimension to define ℓ -Bettinumbers of equivalence relations, and more general measured groupoids, providing an alternativeapproach to Gaboriau’s results. In [127] Sauer and Thom develop further homological tools (includ-ing a spectral sequence for associated to strongly normal subrelations) to study ℓ -Betti numbers forgroups, relations, and measured groupoids.4.2.6. Outer automorphism group. Given an equivalence relation R on ( X , m ) define the corresponding automorphism group asthe group of self isomorphisms:Aut ( R ) = { T ∈ Aut ( X , m ) : T × T ( R ) = R (modulo null sets) } . The subgroup Inn ( R ) of inner automorphisms isInn ( R ) = { T ∈ Aut ( X , m ) : ( x , T ( x )) ∈ R for a.e. x ∈ X } SURVEY OF MEASURED GROUP THEORY 35 This is just the full group [ R ] , but the above notation emphasizes the fact that it is normal in Aut ( R ) and suggest to consider the outer automorphism group Out ( R ) = Aut ( R ) / Inn ( R ) . One might think of Out ( R ) as the group of all measurable permutations of the R -classes on X .Recall (Lemma 4.2) that Inn ( R ) is a huge group as it acts transitively on (classes mod null sets of)measurable subsets of any given size in X . Yet the quotient Out ( R ) might be small (even finite ortrivial), and can sometimes be explicitly computed. Remark 4.13. As an abstract group H = Inn ( R ) is simple, and its automorphisms come fromautomorphisms of R ; in particular Out ( H ) = Out ( R ) . Moreover, Dye’s reconstruction theoremstates that (the isomorphism type of) R is determined by the structure of Inn ( R ) as an abstractgroup (see [80, § I.4] for proofs and further facts).Let us also note that the operation of restriction/amplification of the relation does not alter theouter automorphism group (cf. [47, Lemma 2.2]):Out ( R t ) ∼ = Out ( R ) ( t ∈ R × + ) . The group Aut ( R ) has a natural structure of a Polish group ([62], [64]). First, recall that if ( Y , n ) is a finite or infinite measure Lebesgue space then Aut ( Y , n ) is a Polish group with respect to the weak topology induced from the weak (=strong) operator topology of the unitary group of L ( Y , n ) .This defines a Polish topology on Aut ( R ) when the latter is viewed as acting on the infinite measurespace ( R , ˜ m ) . However, Inn ( R ) is not always closed in Aut ( R ) , so the topology on Out ( R ) mightbe complicated. Alexander Kechris recently found the following surprising connection: Theorem 4.14 (Kechris [80, Theorem 8.1]) . If Out ( R ) fails to be a Polish group, then cost ( R ) = . Now assume that R can be presented as the orbit relation of an essentially free action G y ( X , m ) ,so Aut ( R ) is the group of self orbit equivalences of G y X . The centralizer Aut G ( X , m ) of G in Aut ( X , m ) embeds in Aut ( R ) , and if G is ICC (i.e., has Infinite Conjugacy Classes) then thequotient map Aut ( R ) out −→ Out ( R ) is injective on Aut G ( X , m ) (cf. [62, Lemma 2.6]). So Out ( R ) has a copy of Aut G ( X , m ) , and the latter might be very big. For example in the Bernoulli action G y ( X , m ) = ( X , m ) G , it contains Aut ( X , m ) acting diagonally on the factors. Yet, if G hasproperty (T) then Aut G ( X , m ) · Inn ( R G y X ) is open in the Polish group Aut ( R G y X ) . In this case theimage of Aut G ( X , m ) has finite or countable index in Out ( R G y X ) . This fact was observed by Gefterand Golodets in [64, § ( R G y X ) one looks at OE-cocycles c T : G × X → G corresponding to el-ements T ∈ Aut ( R G y X ) . It is not difficult to see that c T is conjugate in G to the identity (i.e., c T ( g , x ) = f ( gx ) − g f ( x ) for f : X → G ) iff T is in Aut G ( X , m ) · Inn ( R ) . Thus, starting from agroup G or action G y X with strong rigidity properties for cocycles, one controls Out ( R G y X ) viaAut G ( X , m ) . This general scheme (somewhat implicitly) is behind the first example of an equiva-lence relation with trivial Out ( R ) constructed by Gefter [61, 62]. Here is a variant of this construc-tion: This topology coincides with the restriction to Aut ( R ) of the uniform topology on Aut ( X , m ) given by the metric d ( T , S ) = m { x ∈ X : T ( x ) = S ( x ) } . On all of Aut ( X , m ) the uniform topology is complete but not separable; but itsrestriction to Aut ( R ) is separable. Theorem 4.15. Let G be a torsion free group with property (T), K a compact connected Lie groupwithout outer automorphisms, and t : G → K a dense imbedding. Let L < K be a closed subgroupand consider the ergodic actions G y ( K , m K ) and G y ( K / L , m K / L ) by left translations. Then Out ( R G y K ) ∼ = K , Out ( R G y K / L ) ∼ = N K ( L ) / L . In particular, taking K = PO n ( R ) and L ∼ = PO n − ( R ) < K to be the stabilizer of a line in R n , thespace K / L is the projective space P n − , and we getOut ( R G y P n − ) = { } for any property (T) dense subgroup G < PO n ( R ) . Such a group G exists iff n ≥ 5, Zimmer [145,Theorem 7]. The preceding discussion, combined with the cocycles superrigidity theorem 5.20below, and an easy observation that Aut G ( K / L , m K / L ) is naturally isomorphic to N K ( L ) / L , providea self contained sketch of the proof of the theorem.In the above statement Out ( K ) is assumed to be trivial and G to be torsion free just to simplify thestatement. However the assumption that K is connected is essential. Indeed, the dense embeddingof G = PSL n ( Z ) in the compact profinite group K = PSL n ( Z p ) where p is a prime, givesOut ( R PSL n ( Z ) y PSL n ( Z p ) ) ∼ = PSL n ( Q p ) ⋊ Z / Z where the Z / g g tr . The inclusion ⊃ was found in [62],and the equality is proved in [47, Theorem 1.6], where many other computations of Out ( R G y X ) arecarried out for actions of lattices in higher rank Lie groups.Finally, we recall that the recent preprint [117] of Popa and Vaes quoted above (Theorem 4.5)shows that an arbitrary totally disconnected lcsc group G can arise as Out ( R G y X ) for an essentiallyfree action of a free group F ¥ .4.2.7. Cohomology. Equivalence relations have groups of cohomology associated to them similar to cohomology ofgroups. These were introduced by Singer [134] and largely emphasized by Feldman and Moore[40]. Given, say a type II , equivalence relation R on ( X , m ) consider R ( n ) = (cid:8) ( x , . . . , x n ) ∈ X n + : ( x i , x i + ) ∈ R (cid:9) equipped with the infinite Lebesgue measure ˜ m ( n ) defined by˜ m ( n ) ( A ) = Z X n ( x , . . . , x n ) : ( x , . . . , x n ) ∈ R ( n ) o d m ( x ) . Take ( R ( ) , ˜ m ) to be ( X , m ) . Note that ( R ( ) , m ( ) ) is just ( R , ˜ m ) from § m is assumed tobe R -invariant, the above formula is invariant under permutations of x , . . . , x n .Fix a Polish Abelian group A written multiplicatively (usually A = T ). The space C n ( R , A ) of n -cochains consists of equivalence classes (modulo ˜ m ( n ) -null sets) of measurable maps R ( n ) → A ,linked by the operators d n : C n ( R , A ) → C n + ( R , A ) d n ( f )( x , . . . , x n + ) = n + (cid:213) i = f ( x , . . . , ˆ x i , . . . , x ) ( − ) i . SURVEY OF MEASURED GROUP THEORY 37 Call Z n ( R ) = Ker ( d n ) the n -cocycles, and B n ( R ) = Im ( d n − ) the n -coboundaries; the cohomologygroups are defined by H n ( R ) = Z n ( R ) / B n ( R ) . In degree n = c : ( R , m ) → A such that c ( x , y ) c ( y , x ) = c ( x , z ) and 1-coboundaries have the form b ( x , y ) = f ( x ) / f ( y ) for some measurable f : X → A .If A is a compact Abelian group, such as T , then C ( R , A ) is a Polish group (with respect toconvergence in measure). Being closed subgroup in C ( R , A ) , the 1-cocycles Z ( R , A ) form aPolish group. Schmidt [129] showed that B ( R , A ) is closed in Z ( R , A ) iff R is strongly ergodic.There are only few cases where H ( R , T ) were computed: C.C. Moore [102] constructed arelation with trivial H ( R , T ) . Gefter [63] considered H ( R G y G , T ) for actions of property (T)group G densely imbedded in a semi-simple Lie group G . More recently Popa and Sasyk [116]studied H ( R G y X , T ) for property (T) groups G with Bernoulli actions ( X , m ) = ( X , m ) G . In bothcases H ( R G y X , T ) is shown to coincide with group of characters Hom ( G , T ) . Higher cohomologygroups remain mysterious.The fact that A is Abelian is essential to the definition of H n ( R , A ) for n > 1. However indegree n = H ( R , L ) as a set for a general target group L . In fact, this notion iscommonly used in this theory under the name of measurable cocycles (see § A and 5.1 below). Forthe definition in terms of equivalence relations let Z ( R , L ) denote the set of all measurable maps(mod ˜ m -null sets) c : ( R , ˜ m ) → L s . t . c ( x , z ) = c ( x , y ) c ( y , z ) and let H ( R , L ) = Z ( R , L ) / ∼ where the equivalence ∼ between c , c ′ ∈ Z ( R , L ) is declared if c ( x , y ) = f ( x ) − c ′ ( x , y ) f ( y ) for some measurable f : ( X , m ) → L .If R = R G y X is the orbit relation of an essentially free action, then Z ( R G y X , L ) coincideswith the set of measurable cocycles a : G × X → L by a ( g , x ) = c ( x , gx ) . Note that Hom ( G , L ) / L maps into H ( R , L ) , via c p ( x , y ) = p ( g ) for the unique g ∈ G with x = gy . The point of cocyclessuperrigidity theorems is to show that under favorable conditions this map is surjective.4.2.8. Free decompositions. Group theoretic notions such as free products, amalgamated products, and HNN-extensions canbe defined in the context of equivalence relations – see Gaboriau [54, Section IV]. For example, aII -relation R is said to split as a free product of sub-relations { R i } i ∈ I , denoted R = ∗ i ∈ I R i , if(1) R is generated by { R i } i ∈ I , i.e., R is the smallest equivalence relation containing the latterfamily;(2) Almost every chain x = x , . . . , x n = y , where x j − = x j , ( x j − , x j ) ∈ R i ( j ) and i ( j + ) = i ( j ) ,has x = y .If S is yet another subrelation, one says that R splits as a free product of R i amalgamated over S , R = ∗ S R i , if in condition (2) one replaces x j − = x j by ( x j − , x j ) S .The obvious example of the situation above is an essentially free action of a free product ofgroups G = G ∗ G (resp. amalgamated product G = G ∗ G G ) on a probability space ( X , m ) ; inthis case the orbit relations R i = R G i y X satisfy R = R ∗ R (resp. R = R ∗ R R ). In order to define the notion of measurability L should have a Borel structure, and better be a Polish group; often itis a discrete countable groups, or a Lie group. Another useful construction (see Ioana, Peterson, Popa [75]) is as follows. Given measure pre-serving (possibly ergodic) relations R , R on a probability space ( X , m ) for T ∈ Aut ( X , m ) considerthe relation generated by R and T ( R ) . It can be shown that for a residual set of T ∈ Aut ( X , m ) the resulting relation is a free product of R and T ( R ) . A similar construction can be carried outfor amalgamated products. Note that in contrast with the category of groups the isomorphism typeof the free product is not determined by the free factors alone.Ioana, Peterson and Popa [75] obtained strong rigidity results for free and amalgamated productsof ergodic measured equivalence relations and II -factors with certain rigidity properties. Here letus describe some results obtained by Alvarez and Gaboriau [7], which are easier to state; they maybe viewed as an analogue of Bass-Serre theory in the context of equivalence relations. Say that aII -relation R freely indecomposable (FI) if R is not a free product of its subrelations. A group G is said to be measurably freely indecomposable (MFI) if all its essentially free action give freelyindecomposable orbit relations. A group may fail to be MFI even if it is freely indecomposablein the group theoretic sense (surface groups provide an example). Not surprisingly, groups withproperty (T) are MFI (cf. Adams Spatzier [6]); but more generally Theorem 4.16 (Alvarez-Gaboriau [7]) . If G is non-amenable and b ( ) ( G ) = then G is MFI. Theorem 4.17 (Alvarez-Gaboriau [7]) . Let I, J be two finite or countable index sets, { G i } i ∈ I and { L j } j ∈ J be two families of MFI groups, G = ∗ i ∈ I G i , L = ∗ j ∈ J L j , and G y ( X , m ) , L y ( Y , n ) beessentially free p.m.p. actions where each G i y ( X , m ) and L j y ( Y , n ) are ergodic. Assume that G y X SOE ∼ L y Y .Then | I | = | J | and there is a bijection q : I → J so that G i y X SOE ∼ L q ( i ) y Y . The assumption that each free factor is ergodic is important here; Alvarez and Gaboriau also givean analysis of the general situation (where this assumption is dropped).4.3. Rigidity of equivalence relations. The close relation between ME and SOE allows to deduce that certain orbit relations R G y X remember the acting group G and the action G y ( X , m ) up to isomorphism, or up to a virtualisomorphism . This slightly technical concept is described in the following: Lemma 4.18. Suppose an ergodic ME-coupling ( W , m ) of G with L corresponds to a SOE betweenergodic actions T : G y ( X , m ) SOE ∼ L y ( Y , n ) . Then the following are equivalent:(1) There exist short exact sequences → G → G → G → , → L → L → L → where G and L are finite, a discrete ( G , L ) -coupling ( W , m ) and an equivariant map F : ( W , m ) → ( W , m ) ;(2) There exist isomorphism between finite index subgroups G > G ∼ = L < L , so that G y X = X / G and L y Y = Y / L are induced from some isomorphic ergodicactions G y X ∼ = L y Y . SURVEY OF MEASURED GROUP THEORY 39 (3) The SOE (or ME) cocycle G × X → L is conjugate in L to a cocycle whose restriction tosome finite index subgroup G is a homomorphism G → L (the image is necessarily of finiteindex). Let us now state two general forms of relation rigidity. Here is one form Theorem 4.19. Let G y ( X , m ) be an ergodic essentially free action of one of the types below, L an arbitrary group and L y ( Y , n ) as essentially free p.m.p. action whose orbit relation R L y Y isweakly isomorphic to R G y X .Then L is commensurable up to finite kernels to G and the actions G y X and L y Y are virtuallyisomorphic; in particular, the SEO-index is necessarily rational.The list of actions G y X with this SOE-rigidity property include:(1) G is a lattice in a connected, center free, simple, Lie group G of higher rank, and G y Xhas no equivariant quotients of the form G y G / G ′ where G ′ < G is a lattice ( [45, TheoremA] );(2) G = G × · · · × G n where n ≥ , G i ∈ C reg , and G i y ( X , m ) are ergodic; in addition assumethat L y ( Y , n ) is mildly mixing (Monod-Shalom [101] );(3) G is a finite index subgroup in a (product of) Mapping Class Groups as in Theorem 3.20(Kida [85] ). (a) For a concrete example for (1)–(3) one might take Bernoulli actions G y ( X , m ) G . In (1) onemight also take SL n ( Z ) y T n or SL n ( Z ) y SL n ( Z p ) with n ≥ 3. In (2) one might look at F n × F m acting on a compact Lie group K , e.g. SO ( R ) , by ( g , h ) : k gkh − where F n , F m are imbeddeddensely in K . (b) In (1) the assumption that there are no G -equivariant quotient maps X → G / G ′ is necessary,since given such a quotient there is a G ′ -action on some ( X ′ , m ′ ) with G ′ y X ′ SOE ∼ G y X . The rigiditystatement in this case is that this is a complete list of groups and their essentially free actions upto virtual isomorphism ([45, Theorem C]). The appearance of these factors has to do with ( G , m G ) appearing as a quotient of a ( G , L ) -coupling. (c) The basic technique for establishing the stated rigidity in cases (1) – (3) is to establish con-dition (1) in Lemma 4.18. This is done by analyzing a self G -coupling of the form W × L ˇ W (where X = W / L and Y = W / G ) and invoking an analogue of the construction in § (d) In all cases one can sharpen the results (eliminate the ”virtual”) by imposing some benignadditional assumptions: rule out torsion in the acting groups, and impose ergodicity for actions offinite index subgroups.The second stronger form of relation rigidity refers to rigidity of relation morphisms which areobtained from G dsc - cocycle superrigid actions discovered by Sorin Popa (see § Theorem 4.20. Let G y ( X , m ) be a mixing G dsc -cocycle superrigid action, such as:(1) A Bernoulli G -action on ( X , m ) G , where G has property (T), or G = G × G with G non-amenable and G being infinite; (2) G y K / L where G → K is a homomorphism with dense image in a simple compact Liegroup K with trivial p ( K ) , L < K is a closed subgroup, and G has (T).Let L be some group with an ergodic essentially free measure preserving action L y ( Y , n ) ,X ′ ⊂ X a positive measure subset and T : X ′ → Y a measurable map with T ∗ m ≺ n and ( x , x ) ∈ R G y X ∩ ( X ′ × X ′ ) = ⇒ ( T ( x ) , T ( x )) ∈ R L y Y . Then there exist • an exact sequence G −→ G r −→ L with finite G and L < L ; • A L -ergodic subset Y ⊂ Y with < n ( Y ) < ¥ ; • Denoting ( X , m ) = ( X , m ) / G and n = n ( Y ) − · n | Y , there is an isomorphism T : ( X , m ) ∼ =( Y , n ) of L -actions.Moreover, m -a.e. T ( x ) and T ( G x ) are in the same L -orbit. A question of Feldman and Moore. Feldman and Moore showed [39] that any countable Borel equivalence relation R can be gen-erated by a Borel action of a countable group. They asked whether one can find a free action ofsome group, so that R -classes would be in one-to-one correspondence with the acting group. Thisquestion was answered in the negative by Adams [3]. In the context of measured relations, say oftype II , the question is whether it is possible to generate R (up to null sets) by an essentially free action of some group. This question was also settled in the negative in [45, Theorem D], using thefollowing basic constructions:(1) Start with an essentially free action G y ( X , m ) which is rigid as in Theorem 4.19 or 4.20,and let R = ( R G y X ) t with an irrational t .(2) Consider a proper imbedding G ֒ → H of higher rank simple Lie groups choose a lattice G < H , say G = SL ( R ) ⊂ H = SL ( R ) with G = SL ( Z ) . Such actions always admit a Borelcross-section X ⊂ H / G for the G -action, equipped with a holonomy invariant probabilitymeasure m . Take R on ( X , m ) to be the relation of being in the same G -orbit.In case (1) one argues as follows: if some group L has an essentially free action L y ( Y , n ) with ( R G y X ) t = R ∼ = R L y Y then the rigidity implies that G and L are commensurable up to finite kernel,and G y X is virtually isomorphic to L y Y . But this would imply that the index t is rational,contrary to the assumption. This strategy can be carried out in other cases of very rigid actions asin [72, 85, 101, 113]. Theorem 5.20 provides an example of this type R = ( R G y K ) t where G is aKazhdan group densely imbedded in a compact connected Lie group K . So the reader has a sketchof the full proof for a II -relation which cannot be generated by an essentially free action of anygroup.Example of type (2) was introduced by Zimmer in [147], where it was proved that the relation R on such a cross-section cannot be essentially freely generated by a group L which admits a linearrepresentation with an infinite image. The linearity assumption was removed in [45]. This exampleis particularly interesting since it cannot be ”repaired” by restriction/amplification; as any R t canbe realized as a cross-section of the same G -flow on H / G . The space ( Y , n ) might be finite or infinite Lebesgue measure space. SURVEY OF MEASURED GROUP THEORY 41 Question (Vershik) . Let R on ( X , m ) be a II -relation which cannot be generated by an essentiallyfree action of a group; and let G y ( X , m ) be some action producing R . One may assume that theaction is faithful, i.e., G → Aut ( X , m ) is an embedding. What can be said about G and the structureof the measurable family { G x } x ∈ X of the stabilizers of points in X ? In [119] Sorin Popa and Stefaan Vaes give an example of a II -relation R (which is a restrictionof the II ¥ -relation R SL ( Z ) y R to a subset A ⊂ R of positive finite measure) which has property (T)but cannot be generated by an action (not necessarily free) of any group with property (T). 5. T ECHNIQUES Superrigidity in semi-simple Lie groups. The term superrigidity refers to a number of phenomena originated and inspired by the followingcelebrated discovery of G. A. Margulis. Theorem 5.1 (Margulis [93]) . Let G and G ′ be (semi-)simple connected real center free Lie groupswithout compact factors with rk ( G ) ≥ , G < G be an irreducible lattice and p : G → G ′ a homo-morphism with p ( G ) being Zariski dense in G ′ and not precompact. Then p extends to a (rational)epimorphism ¯ p : G → G ′ . The actual result is more general than stated, as it applies to products of semi-simple algebraicgroups over general local fields. We refer the reader to the comprehensive monograph (Margulis[95]) for the general statements, proofs and further results and applications, including the famousArithmeticity Theorem.The core of (some of the available) proofs of Margulis’ superrigidity Theorem is a combinationof the theory of algebraic groups and purely ergodic-theoretic arguments. The result applies touniform and non-uniform lattices alike, it also covers irreducible lattices in higher rank Lie groups,such as SL ( R ) × SL ( R ) . Let us also note that the assumption that p ( G ) is not precompact in G ′ is redundant if p ( G ) is Zariski dense in a real Lie group G ′ (since compact groups over R arealgebraic), but is important in general (cf. SL n ( Z ) < SL n ( Q p ) is Zariski dense but precompact).In [139] R. J. Zimmer has obtained a far reaching generalization of Margulis’ superrigidity,passing from the context of representations of lattices to the framework of measurable cocyclesover probability measure preserving actions (representations of ”virtual subgroups” in Mackey’sterminology). The connection can be briefly summarized as follows: given a transitive action G y X = G / G and some topological group H ; there is a bijection between measurable cocycles G × G / G → H modulo cocycle conjugation and homomorphisms G → H modulo conjugation in HH ( G y G / G , H ) ∼ = Hom ( G , H ) / H (see § A.1, and [52, 146]). In this correspondence, a representation p : G → H extends to a homomor-phism G → H iff the corresponding cocycle p ◦ c : G × G / G → H is conjugate to a homomorphism G → H . Zimmer’s Cocycle Superrigidity Theorem states that under appropriate non-degeneracyassumptions a measurable cocycle over an arbitrary p.m.p. ergodic action G y ( X , m ) is conjugateto a homomorphism. Theorem 5.2 (Zimmer [139], see also [146]) . Let G, G ′ be semi-simple Lie group as in The-orem 5.1, in particular rk R ( G ) ≥ , let G y ( X , m ) be an irreducible probability measure pre-serving action and c : G × X → G ′ be a measurable cocycle which is Zariski dense and not com-pact. Then there exist a (rational) epimorphism p : G → G ′ and a measurable f : X → G ′ so thatc ( g , x ) = f ( gx ) − p ( g ) f ( x ) . In the above statement irreducibility of G y ( X , m ) means mere ergodicity if G is a simplegroup, and ergodicity of the action G i y ( X , m ) for each factor G i in the case of a semi-simplegroup G = (cid:213) ni = G i with n ≥ G < G = (cid:213) G i in a semi-simple group thetransitive action G y G / G is irreducible precisely iff G is an irreducible lattice in G . The notions ofbeing Zariski dense (resp. not compact ) for a cocycle c : G × X → H mean that c is not conjugateto a cocycle c f taking values in a proper algebraic (resp. compact) subgroup of H . SURVEY OF MEASURED GROUP THEORY 43 The setting of cocycles over p.m.p. actions adds a great deal of generality to the superrigidityphenomena. First illustration of this is the fact that once cocycle superrigidity is known for actionsof G it passes to actions of lattices in G : given an action G y ( X , m ) of a lattice G < G one obtainsa G -action on ¯ X = G × G X by acting on the first coordinate (just like the composition operation ofME-coupling § c : G × X → H has a natural lift to ¯ c : G × ¯ X → H and its cohomologyis directly related to that of the original cocycle. So cocycle superrigidity theorems have an almostautomatic bootstrap from lcsc groups to their lattices. The induced action G y ¯ X is ergodic iff G y X is ergodic; however irreducibility is more subtle. Yet, if G y ( X , m ) is mixing then G y ¯ X is mixing and therefore is irreducible.Theorem 3.14 was the first application of Zimmer’s cocycle superrigidity 5.2 (see [139]). Indeed,if a : G × X → G ′ is the rearrangement cocycle associated to an Orbit Equivalence T : G y ( X , m ) OE ∼ G ′ y ( X ′ , m ′ ) where G < G , G ′ < G ′ are lattices, then, viewing a as taking values in G ′ , Zimmerobserves that a is Zariski dense using a form of Borel’s density theorem and deduces that G ∼ = G ′ (here for simplicity the ambient groups are assumed to be simple, connected, center-free andrk R ( G ) ≥ p : G → G ′ and f : X → G ′ so that a ( g , x ) = f ( g x ) p ( g ) f ( x ) − with p : G → p ( G ) < G ′ being isomorphism of lattices. Remark 5.3. At this point it is not clear whether p ( G ) should be (conjugate to) G ′ , and even as-suming p ( G ) = G ′ whether f takes values in G ′ . In fact, the self orbit equivalence of the G action on G / G given by g G g − G gives a rearrangement cocycle c : G × G / G → G which is conjugate to theidentity G → G by a unique map f : G / G → G with f ∗ ( m G / G ) ≺ m G . However, if p ( G ) = G ′ and f takes values in G ′ it follows that the original actions G y ( X , m ) and G ′ y ( X ′ , m ′ ) are isomorphicvia the identification p : G ∼ = G ′ . We return to this point below.5.1.1. Superrigidity and ME-couplings. Zimmer’s cocycle superrigidity theorem applied to OE or ME-cocycles (see § A.2, A.3) has anatural interpretation in terms of ME-couplings. Let G be a higher rank simple Lie group (hereafterimplicitly, connected, and center free), denote by i : G → Aut ( G ) the adjoint homomorphism (whichis an embedding since G is center free). Theorem 5.4 ([44, Theorem 4.1]) . Let G be a higher rank simple Lie group, G , G < G lattices, and ( W , m ) an ergodic ( G , G ) -coupling. Then there exists a unique measurable map F : W → Aut ( G ) so that m-a.e. on WF ( g w ) = i ( g ) F ( w ) , F ( g w ) = F ( w ) i ( g ) − ( g i ∈ G i ) . Moreover, F ∗ m is either the Haar measure on a group G ∼ = Ad ( G ) ≤ G ′ ≤ Aut ( G ) , or is atomic inwhich case G and G are commensurable.Sketch of the proof. To construct such a F , choose a fundamental domain X ⊂ W for the G -actionand look at the ME-cocycle c : G × X → G < G . Apply Zimmer’s cocycle superrigidity theoremto find p : G → G and f : X → G . Viewing G as a subgroup in Aut ( G ) , one may adjust p and f : X → Aut ( G ) by some a ∈ Aut ( G ) , so that p is the isomorphism i : G → G , to get c ( g , x ) = f ( g . x ) − i ( g ) f ( x ) . Define F : W → Aut ( G ) by F ( g x ) = f ( x ) i ( g ) − and check that it satisfies the required relation. Toidentify the measure F ∗ m on Aut ( G ) one uses Ratner’s theorem, which provides the classificationof G -ergodic finite measures on ¯ G / G . (cid:3) Theorem 3.16 is then proved using this fact with G = G plugged into the construction in 5.5which describes an unknown group L essentially as a lattice in G .Note that there are two distinct cases in Theorem 5.4: either F ∗ m is atomic, in which case ( W , m ) has a discrete ME-coupling as a quotient, or F ∗ m is a Haar measure on a Lie group. The former caseleads to a virtual isomorphism between the groups and the actions (this is case (1) in Theorem 4.19);in the latter G y X ∼ = W / G has a quotient of the form G y ¯ G / G (which is [45, Theorem C]).This dichotomy clarifies the situation in Remark 5.3 above.5.2. Superrigidity for product groups. Let us now turn to a brief discussion of Monod-Shalom rigidity (see §§ G ′ has rank one(say G ′ = PSL ( R ) ), while G has higher rank. The conclusion of the superrigidity Theorems 5.1(resp. 5.2) is that either a representation (resp. cocycle) is degenerate , or there is an epimorphism p : G → G ′ . The latter case occurs if and only if G is semi-simple G = (cid:213) G i , with one of the factors G i ≃ G ′ , and p : G → G ′ factoring through the projection p : G pr i −→ G i ≃ G ′ . In this case the givenrepresentation of the lattice extends to p (resp. the cocycle is conjugate to the epimorphism p ).This special case of Margulis-Zimmer superrigidity, i.e., from higher rank G to rank one G ′ ,was generalized by a number of authors [5, 6, 22] replacing the assumption that the target group G ′ has rank one, by more geometric notions, such as G ′ = Isom ( X ) where X is a proper CAT(-1)space. In the setting considered by Monod and Shalom the target group is ”hyperbilic-like” in avery general way, while the source group G rather than being higher rank semi-simple Lie group, isjust a product G = G × · · · × G n of n ≥ arbitrary compactly generated (in fact, just lcsc) groups.The philosophy is that the number n ≥ higher rank propertiesfor such statements. Theorem 5.5 (Monod - Shalom [100]) . Let G = G × · · · × G n be a product of n ≥ lcsc groups,G y ( X , m ) an irreducible p.m.p. action, H is hyperbolic-like group, and c : G × X → H is anon-elementary measurable cocycle.Then there is a non-elementary closed subgroup H < H, a compact normal subgroup K ⊳ H , ameasurable f : X → H, and a homomorphism r : G i → H / K from one of the factors G i of G, sothat the conjugate cocycle c f takes values in H , and G × X → H → H / K is the homomorphism p : G pr i −→ G i r −→ H / K. This beautiful theorem is proved using the technology of second bounded cohomology (developedin [20, 21, 97] and applied in this setting in [96, 100]) with the notions of hyperbolic-like and non-elementary interpreted in the context of the class C reg .Suppose G = G × · · · × G n , n ≥ 2, is a product of ”hyperbolic-like” groups. Let ( W , m ) be a selfME-coupling of G . Consider a ME-cocycle G × X → G which can be viewed as a combination of n Here precompact, or contain in a parabilic SURVEY OF MEASURED GROUP THEORY 45 cocycles c i : G × X c −→ G pr i −→ G i ( i = , . . . , n ) and assume that G y W / G is an irreducible action. Viewing the source group G as a product of n ≥ G i are ”hyperbolic-like” Monod andShalom apply Theorem 5.5. The cocycles arising from ME coupling turn out to be non-elementary,leading to the conclusion that each cocycle c i is conjugate to a homomorphism r i : G j ( i ) → G ′ i ,modulo some reductions and finite kernels,. Since G i commute, the conjugations can be performedindependently and simultaneously on all the cocycles c i . After some intricate analysis of the map i → j ( i ) , kernels and co-kernels of r i , Monod and Shalom show that in the setting of ME couplingsas above the map i → j ( i ) is a permutation and r i are isomorphisms. Thus the original cocycle c can be conjugate to an automorphism of G .This ME-cocycle superrigidity can now be plugged into an analogue of Theorem 5.4 to givea measurable bi- G -equivariant map W → G , which can be used as an input to a construction likeTheorem 5.13. This allows to identify unknown groups L Measure Equivalent to G = G × · · · × G n .The only delicate point is that starting from a G y X SOE ∼ L y Y and the corresponding ( G , L ) -coupling W one needs to look at the self G -coupling S = W × L ˇ W and apply the cocycle superrigidityresult to G y S / G . In order to guarantee that the latter action is irreducible, Monod and Shalomrequire G y X to be irreducible and L y Y to be mildly mixing . They also show that the assumptionon mild mixing is necessary for the result.In [12] Uri Bader and the author proposed to study higher rank superrigidity phenomena using anotion of a (generalized) Weyl group, which works well for higher rank simple Lie groups, arbitraryproducts G = G × · · · × G n of n ≥ A groups, which are close relatives tolattices in SL ( Q p ) . In particular: Theorem 5.6 (Bader - Furman [12]) . Theorem 5.5 holds for target groups from class D ea . Here D ea is a class of hyperbolic-like groups which includes many of the examples in C reg .Plugging this into Monod-Shalom machine one obtains the same results of products of groups inclass D ea .5.3. Strong rigidity for cocycles. In the proof of Theorem 5.4 Zimmer’s cocycle superrigidity was applied to a Measure Equiva-lence cocycle. This is a rather special class of cocycles (see § A.3). If cocycles are analogous torepresentations of lattices then ME-cocycles are analogous to isomorphisms between lattices, inparticular, they have an ”inverse”. Kida’s work on ME for Mapping Class Groups focuses on rigid-ity results for such cocycles. We shall not attempt to explain the ingredients used in this work, butwill just formulate the main technical result analogous to Theorem 5.4. Let G be a subgroup of finiteindex in G ( S g , p ) ⋄ with 3 g + p − > C = C ( S g , p ) denote its curve complex, and Aut ( C ) the groupof its automorphisms; this is a countable group commensurable to G . Theorem 5.7 (Kida [86]) . Let ( W , m ) be a self ME-coupling of G . Then there exists a measurablemap G × G -equivariant map F : W → Aut ( C ) . Returning to the point that ME-cocycles are analogous to isomorphism between lattices, onemight wonder whether Theorem 5.4 holds in cases where Mostow rigidity applies, specifically for G of rank one with PSL ( R ) excluded. In [13] this is proved for G ≃ Isom ( H n R ) , n ≥ 3, and arestricted ME. Theorem 5.8 (Bader - Furman - Sauer [13]) . Theorem 5.4 applies to ℓ -ME-couplings of lattices inG = SO n , ( R ) , n ≥ . The proof of this result uses homological methods ( ℓ and other completions of the usual homol-ogy) combined with a version of Gromov-Thurston proof of Mostow rigidity (for Isom ( H n R ) , n ≥ Cocycle superrigid actions. In all the previous examples the structure of the acting group was the sole source for (su-per)rigidity. Recently Sorin Popa has developed a number of remarkable cocycle superrigidityresults of a completely different nature [109–115]. These results exhibit an extreme form of cocyclesuperrigidity, and rather than relying only on the properties of the acting group G take the advantageof the action G y ( X , m ) . Definition 5.9. An action G y ( X , m ) is C - cocycle superrigid , where C is some class of topolog-ical groups, if for every L ∈ C every measurable cocycle c : G y X → L has the form c ( g , x ) = f ( gx ) − r ( g ) f ( x ) for some homomorphism r : G → L and some measurable f : X → L .Here we shall focus on the class G dsc of all countable groups; however the following resultshold for all cocycles taking values in a broader class U fin which contains G dsc and G cpt – separablecompact groups. Note that the concept of G dsc -cocycle superrigidity is unprecedentedly strong:there is no assumption on the cocycle, the assumption on the target group is extremely weak, the”untwisting” takes place in the same target group. Theorem 5.10 (Popa [113]) . Let G be a group with property (T), G y ( X , m ) = ( X , m ) G be theBernoulli action. Then G y ( X , m ) is G dsc -cocycle superrigid. In fact, the result is stronger: it suffices to assume that G has relative property (T) with respectto a w-normal subgroup G , and G y ( X , m ) has a relatively weakly mixing extension G y ( ¯ X , ¯ m ) which is s-malleable , while G y ( ¯ X , ¯ m ) is weakly mixing. Under these conditions G y ( X , m ) is U fin -cocycle superrigid. See [113] and [48] for the relevant definitions and more details. Weindicate the proof (of the special case above) in § rigidity provided by the acting group and deformations suppliedby the action. In the following remarkable result, Popa further relaxed the property (T) assumption. Theorem 5.11 (Popa [115]) . Let G be a group containing a product G × G where G is non-amenable, G is infinite, and G × G is w-normal in G . Then any Bernoulli action G y ( X , m ) is U fin -cocycle superrigid. The deformations alluded above take place for the diagonal G -action on the square ( X × X , m × m ) .This action is supposed to be ergodic, equivalently the original action should be weak mixing mixing and satisfy addition properties. Isometric actions or, staying in the ergodic-theoretic terminology,actions with discrete spectrum , provide the opposite type of dynamics. These actions have the form G y K / L where L < K are compact groups, G → K a homomorphism with dense image, and G acts by left translations. Totally disconnected K corresponds to profinite completion lim ←− G / G n with SURVEY OF MEASURED GROUP THEORY 47 respect to a chain of normal subgroups of finite index. Isometric actions G y K / L with profinite K , can be called profinite ergodic actions of G – these are precisely inverse limits X = lim ←− X n of transitive G -actions on finite spaces. Adrian Ioana found the following ”virtually G dsc -cocyclesuperrigidity” phenomenon for profinite actions of Kazhdan groups. Theorem 5.12 (Ioana [72]) . Let G y X = K / L be an ergodic profinite action. Assume that G hasproperty (T), or a relative property (T) with respect to a normal subgroup G which acts ergodicallyon X . Then any measurable cocycle c : G y X → L into a discrete group, is conjugate to a cocyclecoming from a finite quotient X → X n , i.e., c is conjugate to a cocycle induced from a homomorphism G n → L of a finite index subgroup. In § Constructing representations. In Geometric Group Theory many QI rigidity results are proved using the following trick. Givena metric space X one declares self-quasi-isometries f , g : X → X to be equivalent ifsup x ∈ X d ( f ( x ) , g ( x )) < ¥ . Then equivalence classes of q.i. form a group , denoted QI ( X ) . This group contains (a quotient of)Isom ( X ) , which can sometimes be identified within QI ( X ) in coarse-geometric terms. If G is a groupwith well understood QI ( G ) and L is an unknown group q.i. to G , then one gets a homomorphism r : L → Isom ( L ) → QI ( L ) ∼ = QI ( G ) whose kernel and image can then be analyzed.Facing a similar problem in the Measure Equivalence category, there is a difficulty in defining ananalogue for QI ( G ) . Let us describe a construction which allows to analyze the class of all groupsME to a given group G from an information about self ME couplings of G .Let G be a lcsc unimodular group. Let us assume that G has the strong ICC property, by whichwe mean that the only regular Borel conjugation invariant probability measure on G is the trivialone, namely the Dirac mass d e at the origin. For countable groups this is equivalent to the conditionthat all non-trivial conjugacy classes are infinite, i.e., the usual ICC property. Connected, (semi)simple Lie groups with trivial center and no compact factors provide other examples of stronglyICC groups.Theorems 5.4, 5.7, 5.8 are instances where a strongly ICC group G has the property that for any ME self coupling ( W , m ) of a lattice G in G there exists a bi- G -equivariant measurable map to G , i.e.a Borel map F : W → G satisfying m -a.e. F (( g , g ) w ) = g F ( w ) g − ( g , g ∈ G ) . It is not difficult to see that the strong ICC property implies that such a map is also unique. (It shouldalso be pointed out that the existence of such maps for self couplings of lattices is equivalent to thesame property for self couplings of the lcsc group G itself; but here we shall stay in the frameworkof countable groups). The following general tool shows how these properties of G can be used toclassify all groups ME to a lattice G < G ; up to finite kernels these turn out to be lattices in G . In the case of G = Isom ( H n ) we restrict to all ℓ -ME couplings. Theorem 5.13 (Bader-Furman-Sauer [13]) . Let G be a strongly ICC lcsc unimodular group, G < Ga lattice, and L some group ME to G and ( W , m ) be a ( G , L ) -coupling. Assume that the self ME-coupling S = W × L ˇ W of G admits a Borel map F : S → G, satisfying a.e. F ([ g x , g y ]) = g · F ([ x , y ]) · g − ( g , g ∈ G ) . Then there exists a short exact sequence K −→ L −→ ¯ L with K finite and ¯ L being a lattice in G, anda Borel map Y : W → G so that a.e. F ([ x , y ]) = Y ( x ) · Y ( y ) − , Y ( g z ) = g · Y ( z ) , Y ( l z ) = Y ( z ) · ¯ l − . Moreover, the pushforward of Y ∗ m is a Radon measure on G invariant under the mapsg g g ¯ l , ( g ∈ G , ¯ l ∈ ¯ L ) . If G is a (semi)-simple Lie group the last condition on the pushforward measure can be analyzedusing Ratner’s theorem (as in Theorem 5.4) to deduce that assuming ergodicity Y ∗ m is either a Haarmeasure on G , or on a coset of its finite index subgroup, or it is (proportional to) counting measureon a coset of a lattice G ′ containing G and a conjugate of ¯ L as finite index subgroups.Theorem 5.13 is a streamlined and improved version of similar statements obtained in [44] forhigher rank lattices, in [101] for products, and in [86] for mapping class groups.5.6. Local rigidity for measurable cocycles. The rigidity vs. deformations approach to rigidity results developed by Sorin Popa led to numberof striking results in von Neumann algebras and in Ergodic theory (some been mentioned in § rigidity side of this approach by the following simple purely ergodic-theoreticstatement, which is a variant of Hjorth’s [68, Lemma 2.5].Recall that one of the several equivalent forms of property (T) is the following statement: a lcscgroup G has (T) if there exist a compact K ⊂ G and e > G -representation p and any ( K , e ) -almost invariant unit vector v there exists a G -invariant unit vector w with k v − w k < / Proposition 5.14. Let G be a group with property (T) and ( K , e ) as above. Then for any ergodicprobability measure preserving action G y ( X , m ) , any countable group L and any pair of cocycles a , b : G × X → L with m { x ∈ X : a ( g , x ) = b ( g , x ) } > − e ( ∀ g ∈ K ) there exists a measurable map f : X → L so that b = a f . Moreover, one can assume that m { x : f ( x ) = e } > / . Proof. Let ˜ X = X × L be equipped with the infinite measure ˜ m = m × m L where m L stands for thecounting measure on L . Then G acts on ( ˜ X , ˜ m ) by g : ( x , l ) ( g . x , a ( g , x ) lb ( g , x ) − ) . SURVEY OF MEASURED GROUP THEORY 49 This action preserves ˜ m and we denote by p the corresponding unitary G -representation on L ( ˜ X , ˜ m ) .The characteristic function v = X ×{ e } satisfies k v − p ( g ) v k = − h p ( g ) v , v i < − ( − e ) = e ( g ∈ G ) and therefore there exists a p ( G ) -invariant unit vector w ∈ L ( ˜ X , ˜ L ) with k v − w k < / 4. Since1 = k w k = R X (cid:229) l | w ( x , l ) | we may define p ( x ) = max l | w ( x , l ) | , L ( x ) = { l : | w ( x , l ) | = p ( x ) } and observe that p ( x ) and the cardinality k ( x ) of the finite set L ( x ) are measurable G -invariantfunctions on ( X , m ) ; hence are a.e. constants p ( x ) = p ∈ ( , ] , k ( x ) = k ∈ { , , . . . } . Since1 / > k v − w k ≥ ( − p ) we have p > / 4. It follows that k = = k w k ≥ k p .Therefore L ( x ) = { f ( x ) } for some measurable map f : X → L . The p ( G ) -invariance of w gives p ( G ) -invariance of the characteristic function of (cid:8) ( x , f ( x )) ∈ ˜ X : x ∈ X (cid:9) , which is equivalent to(5.1) f ( gx ) = a ( g , x ) f ( x ) b ( g , x ) − and b = a f . Let A = f − ( { e } ) and a = m ( A ) . Since (cid:229) l | w ( x , l ) | is a G -invariant function it is a.e. constant k w k = 1. Hence for x / ∈ A we have | w ( x , e ) | ≤ − | w ( x , f ( x )) | = − p , and116 > k v − w k ≥ a · ( − p ) + ( − a ) · ( − ( − p )) ≥ ( − a ) · p > ( − a ) . Thus a = m { x ∈ X : f ( x ) = e } > / > / (cid:3) Cohomology of cocycles. Let us fix two groups G and L . There is no real assumption on G , it may be any lcsc group, butwe shall impose an assumption on L . One might focus on the case where L is a countable group(class G dsc ), but versions of the statements below would apply also to separable compact groups,or groups in a larger class U fin of all Polish groups which imbed in the unitary group of a von-Neumann algebra with finite faithful trace , or a potentially even larger class G binv of groups witha bi-invariant metric, and the class G alg of connected algebraic groups over local fields, say of zerocharacteristic.Given a (not necessarily free) p.m.p. action G y ( X , m ) let Z ( X , L ) , or Z ( G y X , L ) , denotethe space of all measurable cocycles c : G × X → L and by H ( X , L ) , or H ( G y X , L ) , the spaceof equivalence classes of cocycles up to conjugation by measurable maps f : X → L . If L ∈ G alg we shall focus on a subset H ss ( X , L ) of (classes of) cocycles whose algebraic hull is connected,semi-simple, center free and has no compact factors.Any G -equivariant quotient map p : X → Y defines a pull-back Z ( Y , L ) → Z ( X , L ) by c p ( g , x ) = c ( g , p ( x )) , which descends to H ( Y , L ) p ∗ −→ H ( X , L ) . Group inclusions i : L < ¯ L , and j : G ′ < G give rise to push-forward maps H ( X , L ) i ∗ −→ H ( X , ¯ L ) , H ( G y X , L ) j ∗ −→ H ( G ′ y X , L ) . This class, introduced by Popa contains both discrete countable groups and separable compact ones. Question. What can be said about these maps of the cohomology ? The discussion here is inspired and informed by Popa’s [113]. In particular, the following state-ments 5.15(2), 5.16, 5.17(1), 5.19 are variations on Popa’s original [113, Lemma 2.11, Proposition3.5, Lemma 3.6, Theorem 3.1]. Working with class G binv makes the proofs more transparent than in U fin – this was done in [48, § Proposition 5.15. Let p : X → Y be a G -equivariant quotient map. ThenH ( Y , L ) p ∗ −→ H ( X , L ) is injective in the following cases:(1) L is discrete and torsion free.(2) L ∈ G binv and p : X → Y is relatively weakly mixing.(3) L ∈ G alg and H ( − , L ) is replaced by H ss ( − , L ) . The notion of relative weakly mixing was introduced independently by Zimmer [137] andFurstenberg [51]: a G -equivariant map p : X → Y is relatively weakly mixing if the G -action onthe fibered product X × Y X is ergodic (or ergodic relatively to Y ); this turns out to be equivalent tothe condition that G y X contains no intermediate isometric extensions of G y Y . Proposition 5.16. Let i : L < ¯ L ∈ G binv be a closed subgroup, and G y ( X , m ) some p.m.p. action.Then H ( X , L ) i ∗ −→ H ( X , ¯ L ) is injective. This useful property fails in G alg setting: if G < G is a lattice in a (semi-) simple Lie group and c : G × G / G → G in the canonical class then viewed as cocycle into G > G , c is conjugate to theidentity imbedding G ∼ = G < G , but as a G -valued cocycle it cannot be ”untwisted”. Proposition 5.17. Let p : X → Y be a quotient map of ergodic actions, and j : G ′ < G be a normal,or sub-normal, or w-normal closed subgroup acting ergodically on X . Assume that either(1) L ∈ G binv and p is relatively weakly mixing, or(2) L ∈ G alg and one considers H ss ( − , L ) .Then H ( G y Y , L ) is the push-out of the rest of the following diagram:H ( G y X , L ) j ∗ / / H ( G ′ y X , L ) H ( G y Y , L ) p ∗ O O j ∗ / / H ( G ′ y Y , L ) p ∗ O O In other words, if the restriction to G ′ y X of a cocycle c : G × X → L is conjugate to one descendingto G ′ × Y → L , then c has a conjugate that descends to G × X → L . The condition G ′ < G is w-normal (weakly normal) means that there exists a well ordered chain G i of subgroups starting from G ′ and ending with G , so that G i ⊳ G i + and for limit ordinals G j = S i < j G i (Popa). SURVEY OF MEASURED GROUP THEORY 51 Let p i : X → Y i is a collection of G -equivariant quotient maps. Then X has a unique G -equivariantquotient p : X → Z = V Y i , which is maximal among all common quotients p i : Y i → Z . Identify-ing G -equivariant quotients with G -equivariant complete sub s -algebras of X , one has p − ( Z ) = T i p − i ( Y i ) ; or in the operator algebra formalism p − ( L ¥ ( Z )) = T i p − i ( L ¥ ( Y i )) . Theorem 5.18. Let p i : X → Y i , ≤ i ≤ n, be a finite collection of G -equivariant quotients, andZ = V ni = Y i . Then H ( Z , L ) is the push-out of H ( Y i , L ) under conditions (1)-(3) of Proposition 5.15:H ( X , L ) H ( Y , L ) p ∗ ooooooooooo · · · H ( Y i , L ) · · · p ∗ i O O H ( Y n , L ) p ∗ n g g OOOOOOOOOOO H ( Z , L ) p ∗ ooooooooooo p ∗ i O O p ∗ n g g OOOOOOOOOOO More precisely, if c i : G × X → L are cocycles (in case (3) assume [ c i ] ∈ H ss ( Y i , L ) ), whose pull-backs c i ( g , p i ( x )) are conjugate over X , then there exists a unique class [ c ] ∈ H ( Z , L ) , so thatc ( g , p i ( y )) ∼ c i ( g , y ) in Z ( Y i , L ) for all ≤ i ≤ n. The proof of this Theorem relies on Proposition 5.15 and contains it as a special case n = p : X → Y is aminimal quotient to which a cocycle or a family of cocycles can descend up to conjugacy, then it is the minimal or characteristic quotient for these cocycles: if they descend to any quotient X → Y ′ then necessarily X → Y ′ → Y . For example if G < G is a higher rank lattice, L a discrete groupand c : G × X → L is an OE (or ME) cocycle, then either c descends to a G -action on a finite set(virtual isomorphism case), or to X p −→ G / L ′ with L ≃ L ′ lattice in G , where p is uniquely defined by c (initial OE or ME).An important special (and motivating) case of Theorem 5.18 is that of X = Y × Y where G y Y is a weakly mixing action. Then the projections p i : X → Y i = Y , i = , 2, give Z = Y ∧ Y = { pt } and H ( G y { pt } , L ) = Hom ( G , L ) . So Corollary 5.19 (Popa [113, Theorem 3.1], see also [48, Theorem 3.4]) . Let G y Y be a weaklymixing action and c : G × Y → L a cocycle into L ∈ G binv . Let X = Y × Y with the diagonal G -action, c , c : G × X → L the cocycles c i ( g , ( y , y )) = c ( g , y i ) . If c ∼ c over X then there existshomomorphism r : G → L and a measurable f : Y → L , so that c ( g , y ) = f ( gy ) − r ( g ) f ( y ) . Proofs of some results. In this section we shall give a relatively self contained proofs of some of the results mentionedabove.5.8.1. Sketch of a proof for Popa’s cocycle superrigidity theorem 5.10. First note that without loss of generality the base space ( X , m ) of the Bernoulli action may beassumed to be non-atomic. Indeed, Proposition 5.15(2) implies that for each of the classes G dsc ⊂ U fin ⊆ G binv the corresponding cocycle superrigidity descends through relatively weakly mixingquotients, and ([ , ] , dx ) G → ( X , m ) G is such.Given any action G y ( X , m ) consider the diagonal G -action on ( X × X , m × m ) and its centralizerAut G ( X × X ) in the Polish group Aut ( X × X , m × m ) . It always contain the flip F : ( x , y ) ( y , x ) .Bernoulli actions G y X = [ , ] G have the special property (called s-malleability by Popa) thatthere is a path p : [ , ] → Aut G ( X × X ) , with p = Id , p = F . Indeed, the diagonal component-wise action of Aut ([ , ] × [ , ]) on X × X = ([ , ] × [ , ]) G em-beds into Aut G ( X × X ) and can be used to connect Id to F .Fix a cocycle c : G y X → L . Consider the two lifts to X × X → X : c i : G y X × X → L , c i ( g , ( x , x )) = c ( g , x i ) , ( i = , ) . Observe that they are connected by the continuous path of cocycles c t ( g , ( x , y )) = c ( g , p t ( x , y )) ,1 ≤ t ≤ 2. Local rigidity 5.14 implies that c and c are conjugate over X × X , and the proof iscompleted invoking Corollary 5.19. Under the weaker assumption of relative property (T) withrespect to a w-normal subgroup, Popa uses Proposition 5.17.5.8.2. A cocycle superrigidity theorem. We state and prove a cocycle superrigidity theorem, in-spired and generalizing Adrian Ioana’s Theorem 5.12. Thus a number of statements (Theorems 4.15,4.20(2), § deformation vs. rigidity approach.Recall that an ergodic p.m.p. action G y ( X , m ) is said to have a discrete spectrum if theKoopman G -representation on L ( X , m ) is a Hilbert sum of finite dimensional subrepresentations.Mackey proved (generalizing Halmos - von Neumann theorem for Z , and using Peter-Weyl ideas)that discrete spectrum action is measurably isomorphic to the isometric G -action on ( K / L , m K / L ) , g : kL t ( g ) kL , where L < K are compact separable groups and t : G → K is a homomorphismwith dense image. Theorem 5.20 (after Ioana’s Theorem 5.12, [72]) . Let G y ( X , m ) be an ergodic p.m.p. actionwith discrete spectrum. Assume that G has property (T), or contains a w-normal subgroup G withproperty (T) acting ergodically on ( X , m ) . Let L be an arbitrary torsion free discrete countablegroup and c : G × X → L be a measurable cocycle.Then there is a finite index subgroup G < G , a G -ergodic component X ⊂ X (of measure m ( X ) = [ G : G ] − ), a homomorphism r : G → L and a measurable map f : X → L , so thatthe conjugate cocycle c f restricted to G y X → L , is the homomorphism r : G → L . The cocyclec f : G × X → L is induced from r . The assumption that L is torsion free is not essential; in general, one might need to lift the actionto a finite cover ˆ X → X via a finite group which imbeds in L . If K is a connected Lie group,then G = G and X = X = K / L . The stated result is deduced from the case where L is trivial, i.e. X = K , using Proposition 5.15(1). We shall make this simplification and assume G has property(T) (the modification for the more general case uses an appropriate version of Proposition 5.14 andProposition 5.16). An appropriate modification of the result handles compact groups as possibletarget group L for the cocycle. SURVEY OF MEASURED GROUP THEORY 53 Proof. The K -action by right translations: t : x xt − , commutes with the G -action on K ; in fact, K is precisely the centralizer of G in Aut ( K , m K ) . This allows us to deform the initial cocycle c : G × X → L be setting c t ( g , x ) = c ( g , xt − ) ( t ∈ K ) . Let F ⊂ G and e > U of e ∈ K for every t ∈ U there is a unique measurable f t : K → L with c t ( g , x ) = c ( g , xt − ) = f t ( gx ) c ( g , x ) f t ( x ) − m { x : f t ( x ) = e } > . Suppose t , s ∈ U and ts ∈ U . Then f ts ( gx ) c ( g , x ) f ts ( x ) − = c ts ( g , x ) = c ( g , xs − t − )= f t ( gxs − ) c ( g , xs − ) f t ( xs − ) − = f t ( gxs − ) f s ( gx ) c ( g , x ) [ f t ( xs − ) f s ( x ) − ] − . This can be rewritten as F ( gx ) = c ( g , x ) F ( x ) c ( g , x ) − , where F ( x ) = f ts ( x ) − f t ( xs − ) f s ( x ) . Since f t , f s , f ts take value e with probability > / 4, it follows that A = F − ( { e } ) has m ( A ) > 0. Theequation implies G -invariance of A . Thus m ( A ) = t , s , ts ∈ U (5.2) f ts ( x ) = f t ( xs − ) f s ( x ) . If K is a totally disconnected group, i.e., a profinite completion of G as in Ioana’s Theorem 5.12,then U contains an open subgroup K < K . In this case one can skip the following paragraph.In general, let V be a symmetric neighborhood of e ∈ K so that V ⊂ U , and let K = S ¥ n = V n .Then K is an open (hence also closed) subgroup of K ; in the connected case K = K . We shallextend the family { f t : K → L } t ∈ V to be defined for all t ∈ K while satisfying (5.2), using ”cocyclecontinuation” procedure akin to analytic continuation. For t , t ′ ∈ K a V - quasi-path p t → t ′ from t to t ′ is a sequences t = t , t , . . . , t n = t ′ where t i ∈ t i − V . Two V -quasi-paths from t to t ′ are V - close if they are within V -neighborhoods from each other. Two V -quasi-paths p t → t ′ and q t → t ′ are V - homotopic if there is a chain p t → t ′ = p ( ) t → t ′ , . . . , p ( k ) t → t ′ = q t → t ′ of V -quasi-paths where p ( i − ) and p ( i ) are V -close, 1 ≤ i ≤ k . Iterating (5.2) one may continue the definition of f · from t to t ′ along a V -quasi-path; the continuation being the same for V -close quasi-paths, and therefore for V -homotopicquasi-paths as well (all from t to t ′ ). The possible ambiguity of this cocycle continuation procedureis encoded in the homotopy group p ( V ) ( K ) consisting of equivalence classes of V -quasi-pathsfrom e → e modulo V -homotopy. We claim that this group is finite. In the case of a connectedLie group K , p ( V ) ( K ) is a quotient of p ( K ) which is finite since K , contaning a dense property(T) group, cannot have torus factors. This covers the general case as well since p ( V ) ( K ) ”feels”only finitely many factors when K is written as an inverse limit of connected Lie groups and finitegroups. Considering the continuations of f · along V -quasi-paths e → e we get a homomorphism p ( V ) ( K ) → L which must be trivial since L was assumed to be torsion free. Therefore we obtain afamily of measurable maps f t : K → L indexed by t ∈ K and still satisfying (5.2).Let G = t − ( K ) . Then the index [ G : G ] = [ K : K ] is finite. We shall focus on the restriction c of c to G y K . Note that (5.2) is a cocycle equation for the simply transitive action of K on itself. It follows by a standard argument that it is a coboundary. Indeed, for a.e. x ∈ K 14 ALEX FURMAN equation (5.2) holds for a.e. t , s ∈ K . In particular, for a.e. t , x ∈ K , using s = x − x , one obtains f tx − x ( x ) = f t ( x ) f x − x ( x ) . This gives f t ( x ) = f ( xt − ) f ( x ) − , where f ( x ) = f x − x ( x ) . Equation c t = c f t translates into the fact that the cocycle c f ( g , x ) = f ( gx ) − c ( g , x ) f ( x ) satisfies fora.e. x , t c f ( g , xt − ) = c f ( g , x ) . Thus c ( g , x ) does not depend on the space variable. Hence it is a homomorphism c f ( g , x ) = r ( g ) . Finally, the fact that c f is induced from c f is straightforward. (cid:3) SURVEY OF MEASURED GROUP THEORY 55 A PPENDIX A. C OCYCLES Let G y ( X , m ) be a measurable, measure-preserving (sometimes just measure class preserving)action of a topological group G on a standard Lebesgue space ( X , m ) , and H be a topological group.A Borel measurable map c : G × X → H forms a cocycle if for every g , g ∈ G for m -a.e. x ∈ X onehas c ( g g , x ) = c ( g , g . x ) · c ( g , x ) If f : X → H is a measurable map and c : G × X → H is a measurable cocycle, define the f -conjugate c f of c to be c f ( g , x ) = f ( g . x ) − c ( g , x ) f ( x ) . It is straightforward to see that c f is also a cocycle. One says that c and c f are (measurably) conjugate , or cohomologous cocycles. The space of all measurable cocycles G × X → L is denoted Z ( G y X , L ) and the space of equivalence classes by H ( G y X , L ) .Cocycles which do not depend on the space variable: c ( g , x ) = c ( g ) are precisely homomorphisms c : G → H . So cocycles may be viewed as generalized homomorphisms. In fact, any group action G y ( X , m ) defines a measured groupoid G with G ( ) = X , and G ( ) = { ( x , gx ) : x ∈ X , g ∈ G } (see [9] for the background). In this context cocycles can be viewed as homomorphisms G → H .If p : ( X , m ) → ( Y , n ) is an equivariant quotient map between G -actions (so p ∗ m = n , and p ◦ g = g for g ∈ G ) then for any target group L any cocycle c : G × Y → L lifts to ¯ c : G × X → L by¯ c ( g , x ) = c ( g , p ( x )) . Moreover, if c ′ = c f ∼ c in Z ( G y Y , L ) then the lifts ¯ c ′ = ¯ c f ◦ p ∼ ¯ c in Z ( G y X , L ) ; so X p −→ Y induces H ( G y X , L ) p ⋄ ←− H ( G y Y , L ) Note that Hom ( G , L ) is Z ( G y { pt } , L ) and classes of cocycles on G × X → L cohomologous tohomomorphisms is precisely the pull back of H ( G y { pt } , L ) .A.1. The canonical class of a lattice, (co-)induction. Let G < G be a lattice in a lcsc group. By definition the transitive G -action on X = G / G has aninvariant Borel regular probability measure m . Let F ⊂ G be a Borel fundamental domain for theright G -action on G (i.e. F is a Borel subset of G set which meats every coset g G precisely once).Fundamental domains correspond to Borel cross-section s : G / G → G of the projection G → G / G .Define: c s : G × G / G → G , by c s ( g , h G ) = s ( gh G ) − g s ( h G ) . Clearly, this is a cocycle (a conjugate of the identity homomorphism G → G ); however c s takesvalues in the subgroup G of G . This cocycle is associated to a choice of the cross-section s (equivalently, the choice of the fundamental domain); starting from another Borel cross-section s ′ : G / G → G results in a cohomologous cocycle: c s ′ = c f s where f : G / G → G is defined by s ( x ) = f ( x ) s ′ ( x ) . Let G be a lattice in G . Then any action G y ( X , m ) gives rise to the induced G -action (a.k.a. suspension ) on ¯ X = G × G X where G -acts on the first coordinate. Equivalently, ¯ X = G / G × X and g : ( g ′ G , x ) ( gg ′ G , c ( g , g ′ G ) x ) where c : G × G / G → G is in the canonical class. Here the G -invariant finite measure ¯ m = m G / G × m is ergodic iff m is G -ergodic. If a : G × X → H is a cocycle, the induced cocycle ¯ a : G × ¯ X → H is given by ¯ a ( g , ( g ′ G , x )) = a ( c ( g , g ′ G ) , x ) . The cohomology of ¯ a is the same as that of a (one relates maps F : ¯ X → H to f : X → H by f ( x ) = F ( e G , x ) taking insteadof e G a generic point in G / G ). In particular, ¯ a is cohomologous to a homomorpism ¯ p : G → H iff a is cohomologous to a homomorphism G → H ; see [146] for details.Cocycles appear quite naturally in a number of situations such as (volume preserving) smoothactions on manifolds, where choosing a measurable trivialization of the tangent bundle, the deriva-tive becomes a matrix valued cocycle. We refer the reader to David Fisher’s survey [42] where thistype of cocycles is extensively discussed in the context of Zimmer’s programme. Here we shall beinterested in a different type of cocycles: ”rearrangement” cocycles associated to Orbit Equivalence,Measure Equivalence etc. as follows.A.2. OE-cocycles. Let G y ( X , m ) and L y ( Y , n ) be two measurable, measure preserving, ergodic actions on prob-ability spaces, and T : ( X , m ) → ( Y , n ) be an Orbit Equivalence. Assume that the L -action is es-sentially free , i.e., for n -.a.e y ∈ Y , the stabilizer L y = { h ∈ L : h . y = y } is trivial. Then for every g ∈ G and m -a.e. x ∈ X , the points T ( g . x ) , T ( x ) ∈ Y lie on the same L -orbit. Let a ( g , x ) ∈ L denotethe (a.e. unique) element of L with T ( g . x ) = a ( g , x ) . T ( x ) Considering x , g . x , g ′ g . x one checks that a is actually a cocycle a : G × X → L . We shall refer tosuch a as the OE-cocycle , or the rearrangement cocycle, corresponding to T .Note that for m -a.e. x , the map a ( − , x ) : G → L is a bijection; it describes how the G -namesof points x ′ ∈ G . x translate into the L -names of y ′ ∈ L . T ( x ) under the map T . The inverse map T − : ( Y , n ) → ( X , m ) defines an OE-cocycle b : L × Y → G which serves as an ”inverse” to a inthe sense that a.e. b ( a ( g , x ) , T ( X )) = g ( g ∈ G ) . A.3. ME-cocycles. Let ( W , m ) be an ME-coupling of two groups G and L and let Y , X ⊂ W be fundamental domainsfor G , L actions respectively. The natural identification W / L ∼ = X , L w L w ∩ X , translates the G -action on W / L to G y X by g : X ∋ x g a ( g , x ) x ∈ X where a ( g , x ) is the unique element in L taking g x ∈ W into X ⊂ W . It is easy to see that a : G × X → L is a cocycle with respect to the above G -action on X which we denote by a dot g · x to distinguishit from the G -action on W . (If G and L are lattices in G then a : G × G / L → L is the restriction ofthe canonical cocycle G × G / L → L ). Similarly we get a cocycle b : L × Y → G . So the ( G , L ) ME-coupling W and a choice of fundamental domains Y ∼ = W / G , X ∼ = W / L define a pair of cocycles(A.1) a : G × W / L → L , b : L × W / G → G Changing the fundamental domains amounts to conjugating the cocycles and vise versa. SURVEY OF MEASURED GROUP THEORY 57 Remark A.1. One can characterize ME-cocycles among all measurable cocycles a : G × X → L as discrete ones with finite covolume . 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