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# Groups, Geometry, and Dynamics

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**Volume 7, Issue 3, 2013, pp. 577–590**

**DOI: 10.4171/GGD/198**

Published online: 2013-08-27

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

Cyril Houdayer^{[1]}, Sorin Popa

^{[2]}and Stefaan Vaes

^{[3]}(1) École Normale Supérieure de Lyon, France

(2) University of California Los Angeles, United States

(3) Katholieke Universiteit Leuven, Belgium

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.

*Keywords: *W*-superrigidity, deformation/rigidity theory, II$_1$ factor, ergodic equivalence relation, amalgamated free product group

Houdayer Cyril, Popa Sorin, Vaes Stefaan: A class of groups for which every action is W$^*$-superrigid. *Groups Geom. Dyn.* 7 (2013), 577-590. doi: 10.4171/GGD/198