We consider amalgamated free product II 1 factors M = M 1 * B M 2 * B ... and use "deformation/rigidity" and "intertwining" techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitary conjugated into one of the M i 's. We apply this to the case M i are w-rigid II 1 factors, with B equal to either C, to a Cartan subalgebra A in M i , or to a regular hyperfinite II 1 subfactor R in M i , to obtain the following type of unique decomposition results, à la Bass-Serre: If M = (N 1 * C N 2 * C ...) t , for some t > 0 and some other similar inclusions of algebras C ⊂ N j then, after a permutation of indices, (Bwe obtain continuously many non stably isomorphic factors M with fundamental group F (M) equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying F (M) = {1} and Out(M) abelian and calculable. Taking B = R, we get examples of factors with F (M) = {1}, Out(M) = K, for any given separable compact abelian group K. Content J.P.

We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely "remembers" the group G. More precisely, if LG is isomorphic to the von Neumann algebra LΛ of an arbitrary countable group Λ, then Λ must be isomorphic to G. This represents the first superrigidity result pertaining to group von Neumann algebras.2010 Mathematics Subject Classification. Primary: 46L36; Secondary: 20E22.

Abstract. Consider a free ergodic measure preserving profinite action Γ X (i.e. an inverse limit of actions Γ X n , with X n finite) of a countable property (T) group Γ (more generally of a group Γ which admits an infinite normal subgroup Γ 0 such that the inclusion Γ 0 ⊂ Γ has relative property (T) and Γ/Γ 0 is finitely generated) on a standard probability space X. We prove that if w : Γ × X → Λ is a measurable cocycle with values in a countable group Λ, then w is cohomologous to a cocycle w ′ which factors through the map Γ × X → Γ × X n , for some n. As a corollary, we show that any orbit equivalence of Γ X with any free ergodic measure preserving action Λ Y comes from a (virtual) conjugacy of actions. §0. Introduction.During the past decade, the orbit equivalence theory of measure preserving actions of groups has been an extremely active area, with many new rigidity results having been proven (see the surveys [Sh1],[P4]). In particular, certain classes of group-actions Γ X have been shown to be orbit equivalent superrigid, i.e. such that the equivalence relation R Γ on X of belonging to the same Γ-orbit (x ∼ y iff Γx = Γy) remembers both the group Γ and the action Γ ]). Since all ergodic actions ΓX of all infinite amenable groups Γ induce isomorphic equivalence relations R Γ (up to a probability space isomorphism) ([Dy], [OW], [CFW]), such a rigidity phenomenon is very surprising and is characteristic to non-amenable groups only.The main purpose of this paper is to present a new class of orbit equivalent superrigid actions. To explain this in more detail, we first review a few concepts, starting with the notion of orbit equivalence. Let Γ X be a free ergodic measure preserving action of a countable group Γ on a standard probability space (X, µ) (i.e. isomorphic with the unit interval with the Lebesgue measure). Given another such action Λ Y , a probability space isomorphism θ : X → Y is called an orbit equivalence (OE) between the actions Γ X and Λ Y , if θ(Γx) = Λθ(x), a.e. x ∈ X. If we moreover have that θΓθ −1 = Λ, then θ is called a conjugacy between Γ X and Λ Y .

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We consider group measure space II 1 _{1} factors M = L ∞ ( X ) ⋊ Γ M=L^{\infty }(X)\rtimes \Gamma arising from Bernoulli actions of ICC property (T) groups Γ \Gamma (more generally, of groups Γ \Gamma containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for ∗ * –homomorphisms θ : M → M ⊗ ¯ M \theta :M\rightarrow M\overline {\otimes }M . We deduce that the action Γ ↷ X \Gamma \curvearrowright X is W ∗ ^{*} –superrigid, i.e. if Λ ↷ Y \Lambda \curvearrowright Y is any free, ergodic, measure preserving action such that the factors M = L ∞ ( X ) ⋊ Γ M=L^{\infty }(X)\rtimes \Gamma and L ∞ ( Y ) ⋊ Λ L^{\infty }(Y)\rtimes \Lambda are isomorphic, then the actions Γ ↷ X \Gamma \curvearrowright X and Λ ↷ Y \Lambda \curvearrowright Y must be conjugate. Moreover, we show that if p ∈ M ∖ { 1 } p\in M\setminus \{1\} is a projection, then p M p pMp does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that Γ \Gamma is torsion free). We also prove a rigidity result for ∗ * –homomorphisms θ : M → M \theta :M\rightarrow M , this time for Γ \Gamma in a larger class of groups than above, now including products of non–amenable groups. For certain groups Γ \Gamma , e.g. Γ = F 2 × F 2 \Gamma =\mathbb {F}_{2}\times \mathbb {F}_{2} , we deduce that M M does not embed into p M p pMp , for any projection p ∈ M ∖ { 1 } p\in M\setminus \{1\} , and obtain a description of the endomorphism semigroup of M M .