A class of groups for which every action is W${}^{\ast }$-superrigid

Abstract

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products $A⋊\Gamma$ covering certain cases where $\Gamma$ is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if $\Sigma <\mathrm{SL}(3,\mathbb{Z})$ denotes the subgroup of matrices $g$ with $g_{31}=g_{32}=0$, then any free ergodic probability measure preserving action of $\Gamma =\mathrm{SL}(3,\mathbb{Z}){\ast }_{\Sigma }\mathrm{SL}(3,\mathbb{Z})$ is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.