Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations

We provide a new large class of countable icc groups $\mathcal{A}$ for which the product rigidity result from [CdSS15] holds: if ${\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_n\in \mathcal{A}$ and $\mathrm{\Lambda }$ is any group such that $L({\mathrm{\Gamma }}_1\times \cdots \times {\mathrm{\Gamma }}_n)\cong L(\mathrm{\Lambda })$, then there exists a product decomposition $\mathrm{\Lambda }={\mathrm{\Lambda }}_1\times \cdots \times {\mathrm{\Lambda }}_n$ such that $L({\mathrm{\Lambda }}_i)$ is stably isomorphic to $L({\mathrm{\Gamma }}_i)$, for any $1\le i\le n$. Class $\mathcal{A}$ consists of groups $\mathrm{\Gamma }$ for which $L(\mathrm{\Gamma })$ admits an s-malleable deformation in the sense of Sorin Popa and it includes all non-amenable groups $\mathrm{\Gamma }$ such that either (a) $\mathrm{\Gamma }$ admits an unbounded 1-cocycle into its left regular representation, or (b) $\mathrm{\Gamma }$ is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W${}^{\ast }$-superrigid groups and new rigidity results in the C${}^{\ast }$-algebra theory.