Amalgamated Free Product Rigidity for Group von Neumann Algebras

We provide a fairly large family of amalgamated free product groups $\mathrm{\Gamma }={\mathrm{\Gamma }}_1{\ast }_{\mathrm{\Sigma }}{\mathrm{\Gamma }}_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that ${\mathrm{\Gamma }}_i$ is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and $\mathrm{\Sigma }$ is icc amenable and has trivial one-sided commensurator in ${\mathrm{\Gamma }}_i$, for every $i\in \{1,2\}$. Then $\mathrm{\Gamma }$ satisfies the following rigidity property: any group $\text{La}$ such that $L(\text{La})$ is isomorphic to $L(\text{G})$ admits an amalgamated free product decomposition $\text{La}={\text{La}}_1{\ast }_{\mathrm{\Delta }}{\text{La}}_2$ such that the inclusions $L(\Delta)\subseteq L(\La_i)$and$L(\Sigma)\subseteq L(\G_i)$areisomorphic,forevery$i\in{1,2}$. 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.