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


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