JournalsggdVol. 7, No. 2pp. 403–417

Rigidity for equivalence relations on homogeneous spaces

  • Adrian Ioana

    University of California, San Diego, United States
  • Yehuda Shalom

    Tel Aviv University, Israel
Rigidity for equivalence relations on homogeneous spaces cover
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Abstract

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices Γ\Gamma and Λ\Lambda in a semisimple Lie group GG with finite center and no compact factors we prove that the action ΓG/Λ\Gamma\curvearrowright G/\Lambda is rigid. If in addition GG has property (T) then we derive that the von Neumann algebra L(G/Λ)ΓL^{\infty}(G/\Lambda)\rtimes\Gamma has property (T). We also show that if the stabilizer of any non-zero point in the Lie algebra of GG under the adjoint action of GG is amenable (e.g., if G=SL2(R)G=\operatorname{SL}_2(\mathbb R)), then any ergodic subequivalence relation of the orbit equivalence relation of the action ΓG/Λ\Gamma\curvearrowright G/\Lambda is either hyperfinite or rigid.

Cite this article

Adrian Ioana, Yehuda Shalom, Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn. 7 (2013), no. 2, pp. 403–417

DOI 10.4171/GGD/187