Some rigidity results for II${}_1$ factors arising from wreath products of property (T) groups

Abstract

We show that any infinite collection $({\Gamma }_n{)}_{n\in \mathbb{N}}$ of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon. If $\Lambda$ is an arbitrary group such that $L({\oplus }_{n\in \mathbb{N}}{\Gamma }_n)\cong L(\Lambda )$ then there exists an infinite direct sum decomposition $\Lambda =({\oplus }_{n\in \mathbb{N}}{\Lambda }_n)\oplus A$ with $A$ icc amenable such that, for all $n\in \mathbb{N}$, up to amplifications, we have $L({\Gamma }_n)\cong L({\Lambda }_n)$ and $L({\oplus }_{k\ge n}{\Gamma }_k)\cong L(({\oplus }_{k\ge n}{\Lambda }_k)\oplus A)$. The result is sharp and complements the previous finite product rigidity property found in [CdSS16]. Using this we provide an uncountable family of restricted wreath products $\Gamma \cong \Sigma \wr \Delta$ of icc, property (T) groups $\Sigma$, $\Delta$ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras $L(\Gamma )$. Along the way we highlight several applications of these results to the study of rigidity in the ${\mathbb{C}}^{\ast }$-algebra setting.

27 pages; first version; comments are welcome!