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

TL;DR

This paper establishes a new infinite product rigidity phenomenon for von Neumann algebras arising from wreath products of property (T) groups, showing that their algebraic structure can be recovered from the von Neumann algebra up to certain decompositions.

Contribution

It introduces an infinite product rigidity result for II$_1$ factors from property (T) groups, extending previous finite product results and enabling recognition of wreath product structures from von Neumann algebras.

Findings

Proves infinite product rigidity for von Neumann algebras of certain property (T) groups.

Identifies conditions under which wreath product structures are recoverable from von Neumann algebras.

Provides applications to rigidity in the $ ext{C}^*$-algebra setting.

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 $L(\oplus_{k\geq n} \Gamma_k )\cong L((\oplus_{k\geq n} \Lambda_k) \oplus A)$. 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 $\mathbb C^*$-algebra setting.