  This paper includes 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
\cite{CS}, 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 II$_1$ factor associated with a weakly amenable group in the class
$\mathcal S$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the
following product version of this result: any maximal abelian
$\star$-subalgebra of any II$_1$ factor associated with a finite product of
weakly amenable groups in the class $\mathcal S$ of Ozawa has an amenable
normalizing algebra. Finally, pairing some of these results with cocycle
superrigidity results from \cite{IoaCSR}, it follows that compact actions by
finite products of lattices in $Sp(n, 1)$, $n \geq2$, are virtually
$W^*$-superrigid.

Operator Algebras;Group Theory