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.

Abstract. We introduce the notion of L 2 -rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L 2 -rigidity passes to normalizers and is satisfied by nonamenable II 1 factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M = LΓ where Γ is a finitely generated group with b (2) 1 (Γ) > 0, then any nonamenable regular subfactor of M is prime and does not have properties Γ or (T). In particular this gives a new approach for showing primeness of all nonamenable subfactors of a free group factor thus recovering a well known recent result of N. Ozawa.

Abstract. In this article we study cocycles of discrete countable groups with values in ℓ 2 G and the ring of affiliated operators U G. We clarify properties of the first cohomology of a group G with coefficients in ℓ 2 G and answer several questions from [14]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first ℓ 2 -Betti number. We give numerous applications and examples of groups which satisfy our assumptions.

Abstract. 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. R. Parthasarathy and 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.

Abstract. Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.