On the structural theory of II1\rm II_1 factors of negatively curved groups

Ozawa showed that for any i.c.c., hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson, Ozawa and Popa, and Ozawa, we strengthen this result by showing that these factors are strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana, we show that profinite actions of lattices in Sp(n,1), n>1, are virtually W*-superrigid.Comment: Fina version; to appear as such in Annales Scientifiques de l'EN

On a Question of D. Shlyakhtenko

In this short note we construct two countable, infinite conjugacy class groups which admit free, ergodic, probability measure preserving orbit equivalent actions, but whose group von Neumann algebras are not (stably) isomorphic

Some unique group-measure space decomposition results

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving action on a probability space G_1 x G_2 on X gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra. Pairing this result with Popa's Orbit Equivalence Superrigidity Theorem we obtain new examples of W*-superrigid actions.Comment: Revised proofs in Section

On Relative Property (T) and Haagerup's Property

We consider the following three properties for countable discrete groups $\Gamma$: (1) $\Gamma$ has an infinite subgroup with relative property (T), (2) the group von Neumann algebra $L\Gamma$ has a diffuse von Neumann subalgebra with relative property (T) and (3) $\Gamma$ does not have Haagerup's property. It is clear that (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3). We prove that both of the converses are false