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We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement are given with fewer assumptions on the defining graphs. We also provide a quantified version of our measure equivalence classification theorem, that keeps track of the integrability of associated cocycles. As an application, we solve an inverse problem in quantitative orbit equivalence for a large family of right-angled Artin groups. We then establish several rigidity theorems. First, in the spirit of work of Monod-Shalom, we achieve rigidity in orbit equivalence for probability measure-preserving actions of graph products, upon imposing extra ergodicity assumptions. Second, we establish a sufficient condition on the defining graph and on the vertex groups ensuring that a graph product G is rigid in measure equivalence among torsion-free groups (in the sense that every torsion-free countable group H which is measure equivalent to G, is in fact isomorphic to G). Using variations over the Higman groups as the vertex groups, we construct the first example of a group which is rigid in measure equivalence, but not in quasi-isometry, among torsion-free groups., Comment: 166 pages, 26 figures

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a "conditional" Pontryagin duality for spaces of abstract measurable maps., Comment: 28 pages [v4]: removed material concerning weak conditional elements, following an error pointed out by the referee. Also modified the notation in line with referee comments on this paper and related papers

We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G., Comment: This new version adds more context in the introduction and fixes minor typos

Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property., Comment: 31 pages, Theorem 1.4 improved

Let us say that a discrete countable group is stable if it has an ergodic, free, probability-measure-preserving and stable action. Let G be a discrete countable group with a central subgroup C. We present a sufficient condition and a necessary condition for G to be stable. We show that if the pair (G, C) does not have property (T), then G is stable. We also show that if the pair (G, C) has property (T) and G is stable, then the quotient group G/C is stable., Comment: 22 pages, 3 figures, minor corrections

We present a general setting to investigate U_fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some U_fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be U_fin-cocycle superrigid., Comment: Generalized Theorem 3.1 and added Theorem 5.3.

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if $G_1,G_2$ are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over $G_1$ is stably orbit equivalent to every nontrivial Bernoulli shift over $G_2$. This answers a question of S. Popa., Comment: The second version has minor notational changes

The following result is proven. Let $G_1 \cc^{T_1} (X_1,\mu_1)$ and $G_2 \cc^{T_2} (X_2,\mu_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let $\Gamma_i = G_i *H$ be the free product. Then the actions of $\Gamma_1$ and $\Gamma_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $\Gamma$ is a free group, then all nontrivial Bernoulli shifts over $\Gamma$ are orbit-equivalent., Comment: New version. The cocycles have been standardized and proofs simplified. A reference has been corrected.

This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors., Comment: 24 pages, continuation of arXiv:0706.3623

Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma\times X\to \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $\Gamma\times X\to \Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.

We prove that if $G$ is a countable discrete group with property (T) over an infinite subgroup $H

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., Comment: v.2: slight changes in the presentation, (some) typos corrected, wq-normal subgroups added. 35 pages

We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L^2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L^2-Betti numbers under orbit equivalence., Comment: 10 pages, no figures

We study the equivalence relation $R_N$ generated by the (non-free) action of the generalized Thompson group $F_N$ on the unit interval. We show that this relation is a standard, quasipreserving ergodic equivalence relation. Using results of Feldman-Moore, Krieger and Connes we prove that the von Neumann algebra $M(R_N)$ associated to $R_N$ is the hyperfinite type $III_{\lambda}$ factor, with $\lambda=1/N$.

We extend the Delorme-Guichardet characterization of Kazhdan property $T$ groups to $r$-discrete measured groupoids. We give several applications, in particular to stability results of Kazhdan property $T$ and to the study of cocycles taking values in a group having the Haagerup property., Comment: 43 pages

We define an analog of Voiculescu's free entropy for n-tuples of unitaries (u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital diffuse abelian subalgebra B in M. Using this quantity, we define the free dimension \delta_{0}(u_{1},..,u_{n}\btw B). This number depends on (u_{1},... ,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an measirable equivalence relation on [0,1] generated by n automorphisms \alpha_{1},...,\alpha_{n}, let u_{1},..., u_{n} be the unitaries implementing \alpha_{1},..,\alpha_{n} in the Feldman-Moore crossed product algebra M=W^{*}([0,1],R), and let B be the canonical copy of L^\infty functions on [0,1] inside M. In this way, we obtain an invariant \delta (R)=\delta _{0}(u_{1},...,u_{n},\btw B) of the equivalence relation (R). If R is treeable, \delta (R) coincides with the cost C(R) of R in the sense of Gaboriau. For a general equivalence relation R posessing a finite graphing, \delta(R)\leq C(R). Using the notion of free dimension, we define an dynamical entropy invariant for an automorphism of a measurable equivalence relation (or more generally of an r-discrete measure groupoid), and give examples.