Amenable absorption in von Neumann algebras of hyperbolic groups

Abstract

We prove that the von Neumann algebra $\cL(G)$ associated with any hyperbolic group $G$ satisfies the following \emph{amenable absorption property}: for any infinite maximal amenable subgroup $H \leqslant G$ and any amenable von Neumann subalgebra $\mathcal{Q} \subset \cL(G)$ with diffuse intersection with $\cL(H)$, one must have $\mathcal{Q} \subset \cL(H)$. This strengthens a result of Boutonnet and Carderi \cite{BC2}. We also establish similar amenable absorption results for the broader class of acylindrically hyperbolic groups, including relatively hyperbolic groups, mapping class groups, and limit groups.