Rigidity for equivalence relations on homogeneous spaces

Details Rigidity for equivalence relations on homogeneous spaces Rigidity for equivalence relations on homogeneous spaces

2010-10-19

arxiv.org/abs/1010.3778v2

Mathematics

Dynamical Systems

Scientific paper

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $\Gamma,\Lambda$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $\Gamma\curvearrowright G/\Lambda$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\infty}(G/\Lambda)\rtimes\Gamma$ has property (T). We also show that if the adjoint action of $G$ on the Lie algebra of $G$ - $\{0\}$ is amenable (e.g. if $G=SL_2(\Bbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $\Gamma\curvearrowright G/\Lambda$ is either hyperfinite or rigid.

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