Amalgamated Free Product Rigidity for Group von Neumann Algebras

Abstract

We provide a fairly large family of amalgamated free product groups $\Gamma ={\Gamma }_1{\ast }_{\Sigma }{\Gamma }_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that ${\Gamma }_i$ is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and $\Sigma$ is icc amenable and has trivial one-sided commensurator in ${\Gamma }_i$, for every $i\in \{1,2\}$. Then $\Gamma$ satisfies the following rigidity property: any group $\La$ such that $L(\La)$ is isomorphic to $L(\G)$ admits an amalgamated free product decomposition $\La=\La_1\ast_Δ\La_2$ such that the inclusions $L(Δ)\subseteq L(\La_i)$ and $L(Σ)\subseteq L(\G_i)$ are isomorphic, for every $i\in \{1,2\}$. This result significantly strengthens some of the previous Bass-Serre rigidity results for von Neumann algebras. As a corollary, we obtain the first examples of amalgamated free product groups which are W${}^{\ast }$-superrigid.

arXiv comment24 pages; this new version includes new examples of C*-superrigid groups (Corollary C) and more references; comments are welcome!