Product rigidity in von Neumann and C$^*$-algebras via s-malleable deformations

Abstract

We provide a new large class of countable icc groups A for which the product rigidity result from [CdSS15] holds: if Γ1, . . . , Γn ∈ A and Λ is any group such that L(Γ1 ו • •×Γn) ∼ = L(Λ), then there exists a product decomposition Λ = Λ1 × • • • × Λn such that L(Λi) is stably isomorphic to L(Γi), for any 1 ≤ i ≤ n. Class A consists of groups Γ for which L(Γ) admits an s-malleable deformation and it includes all non-amenable groups Γ such that either (a) Γ admits an unbounded 1-cocycle into its left regular representation, or (b) Γ is an arbitrary wreath product group with amenable base. As a byproduct of these results, we obtain new examples of W * -superrigid groups and new rigidity results in the C * -algebra theory.