Abstract. We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a measure preserving action Γ � X of Γ on a probability space (X, µ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action Γ � [0,1] Γ) is cohomologous to a group morphism of Γ into V. We use the case V discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between Γ � X and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. There has recently been increasing interest in the study of measure preserving actions of groups on (non-atomic) probability spaces up to orbit equivalence (OE), i.e. up to isomorphisms of probability spaces taking the orbits of one action onto the orbits of

Abstract. We prove that if a countable group Γ contains a non-amenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is non-amenable and has infinite center or is a product of infinite groups) then any measure preserving Γ-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L ∞ X ⋊Γ ≃ L ∞ Y ⋊Λ comes from a conjugacy of the actions. 1.

Measure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups.

Abstract. For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [MSb] hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces. 1.

Abstract. We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E ∞ is not essentially free. 1.

Abstract. These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of “rigid ” groups) satisfy the assumptions of Popa’s theorem. 1. Introduction and Statement

Abstract. Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ � [0, 1] Γ, where [0,1] Γ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation R of S, there exists a partition {Xi} i≥0 of [0, 1] Γ with R-invariant measurable sets such that R |X0 is hyperfinite and R |Xi is strongly ergodic (hence ergodic), for every i ≥ 1. §1. Introduction and statement of results. During the past decade there have been many interesting new directions arising in the field of measurable group theory. One direction came from the deformation/rigidity theory developed recently by S. Popa in order to study group actions and von Neumann algebras ([P5]). Using this theory, Popa obtained striking rigidity

Abstract. Let G be a countable group. We proof that there is a model companion for the approximate theory of a Hilbert space with a group G of automorphisms. We show that G is amenable if and only if the structure induced by countable copies of the regular representation of G is existentially closed. 1.

Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on M X, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ � M X has almost invariant sets. 1.

Abstract. The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.