Rigidity results for wreath product II$_1$ factors

TL;DR

This paper establishes rigidity results for certain II$_1$ factors formed by wreath products, showing that isomorphisms imply conjugacy of actions and allowing distinctions between related group von Neumann algebras.

Contribution

It proves that isomorphism of these factors implies cocycle conjugacy of Bernoulli shift actions, revealing new rigidity phenomena for wreath product II$_1$ factors.

Findings

Isomorphism implies cocycle conjugacy of Bernoulli shifts

Groups acting are necessarily isomorphic

Distinguishes classes of group von Neumann algebras

Abstract

We consider II$_1$ factors of the form $M=\bar{\bigotimes}_{G}N\rtimes G$, 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 $\bar{\bigotimes}_{G}N$ 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.