# Amalgamated Free Product Rigidity for Group von Neumann Algebras

## TL;DR

This paper establishes a rigidity result for von Neumann algebras arising from certain amalgamated free product groups, enabling their structure to be fully recovered from the algebra and providing new examples of W*-superrigid groups.

## Contribution

It proves a new rigidity theorem for von Neumann algebras of amalgamated free product groups with specific properties, strengthening previous Bass-Serre rigidity results.

## Key findings

- Von Neumann algebra isomorphisms imply group decomposition preservation
- First examples of W*-superrigid amalgamated free product groups
- Enhanced understanding of group von Neumann algebra rigidity

## 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 $\La=\La_1\ast_\Delta \La_2$ such that the inclusions $L(\Delta)\subseteq L(\La_i)$ and $L(\Sigma)\subseteq L(\G_i)$ 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$^*$-superrigid.