Abstract. We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M, β HT n (M), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (Mt) = β HT n (M)/t, ∀t> 0, and a Künneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 ⋊ SL(2, Z)), for which β HT 1 (M) = β1(SL(2, Z)) = 1/12. Thus, Mt ̸ ≃ M, ∀t ̸ = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.

Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1

Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X,µ), and let RΓ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation RΓ on X determines the group Γ and the action (X,µ,Γ) uniquely, up to finite groups. The natural action of SLn(Z) on the n-torus R n /Z n, for n> 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II1, which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.

Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications

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. Let R a be countable ergodic equivalence relation of type II1 on a standard probability space (X, µ). The group Out R of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map R-classes to R-classes modulo those which preserve almost every R-class. We analyze the group Out R for relations R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of Out R and explicitly computing Out R for the standard actions. The method is based on Zimmer’s superrigidity for measurable cocycles, Ratner’s theorem and Gromov’s Measure Equivalence construction. 1. Introduction and Statement

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.

. Let \Gamma be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which \Gamma can be embedded as a lattice. For lattices \Gamma in rank one groups G (with the only exception of non-uniform lattices in G ' SL 2 (R), which are virtually free groups) we give a similar description of all possible locally compact groups H, in which \Gamma can be embedded as a uniform lattice. 1. Introduction and Statement of the Main Results Throughout this paper we use the following terminology: (semi)simple Lie group stands for (semi)simple, connected real Lie group with finite center and no non-trivial compact factors. If (semi)simple Lie groups G, G 0 are locally isomorphic we write G ' G 0 . Locally compact groups are assumed to be second countable, but otherwise may be very general. A countable subgroup \Gamma in a locally compact group G is said to form a lattice if it is discrete and G=\Gamma carries a finite G-invariant measure ...

Abstract. We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice embeddings of the mapping class group into a locally compact second countable group. We also obtain similar results for finite direct products of mapping class groups. 1.

Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank n are incomparable with respect to Borel reducibility.