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

By Cyril Houdayer, Sorin Popa, S. Vaes. 2010

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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 < SL(3,\Z) denotes the subgroup of matrices g with g_31 = g_32 = 0, then any free ergodic probability measure preserving action of \Gamma = SL(3,\Z) *_\Sigma SL(3,\Z) is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/514fd0619a9ea473acdfd9daf792c6c78ae7b469