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

### Cyril Houdayer

École Normale Supérieure de Lyon, France### Sorin Popa

University of California Los Angeles, United States### Stefaan Vaes

Katholieke Universiteit Leuven, Belgium

## 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.

## Cite this article

Cyril Houdayer, Sorin Popa, Stefaan Vaes, A class of groups for which every action is W$^*$-superrigid. Groups Geom. Dyn. 7 (2013), no. 3, pp. 577–590

DOI 10.4171/GGD/198