W$^*$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups

TL;DR

This paper proves that for products of hyperbolic groups, the von Neumann algebra structure uniquely determines the number of factors, strengthening previous prime decomposition results and providing new examples of non-stably equivalent lattices.

Contribution

It establishes a $W^*$-rigidity result for products of hyperbolic groups, removing assumptions on the group $\Lambda$ and strengthening prime decomposition theorems.

Findings

Unique prime decomposition for von Neumann algebras of hyperbolic group products

Examples of lattices with non-stably equivalent II$_1$ factors

Extension of Ozawa and Popa's rigidity results

Abstract

We show that if $\Gamma = \Gamma_1\times\dotsb\times \Gamma_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $\Lambda$ which is $W^*$-equivalent to $\Gamma$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $\Lambda$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors.