On the structural theory of II1 factors of negatively curved groups, II: Actions by product groups

This paper contains a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance, we show that every II1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid. There is also the following product version of this result: any maximal abelian ⋆-subalgebra of any II1 factor associated with a finite product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with a cocycle superrigidity result of Ioana, it follows that compact actions by finite products of lattices in Sp(n,1), n≥2, are virtually W∗-superrigid.