We prove the uniqueness of the group measure space Cartan subalgebra in crossed products covering certain cases where is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if denotes the subgroup of matrices with , then any free ergodic probability measure preserving action of is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.
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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. 577–590DOI 10.4171/GGD/198