Rigidity results for wreath product II${}_1$ factors

Abstract

We consider II${}_1$ factors of the form $M={\overline{⨂}}_{G}N⋊G$, where either i) $N$ is a non-hyperfinite II${}_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II${}_1$ factor and $G$ is ICC group and where $G$ acts on ${\overline{⨂}}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting are isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.

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