A class of groups for which every action is W$^*$-superrigid

Abstract

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes \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})*_\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.