JournalsggdVol. 7, No. 3pp. 577–590

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

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products AΓA \rtimes \Gamma covering certain cases where Γ\Gamma is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if Σ<SL(3,Z)\Sigma <\mathrm{SL}(3,\mathbb{Z}) denotes the subgroup of matrices gg with g31=g32=0g_{31} = g_{32}=0, then any free ergodic probability measure preserving action of Γ=SL(3,Z)ΣSL(3,Z)\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