# Groups, Geometry, and Dynamics

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**Volume 7, Issue 2, 2013, pp. 403–417**

**DOI: 10.4171/GGD/187**

Published online: 2013-05-07

Rigidity for equivalence relations on homogeneous spaces

Adrian Ioana^{[1]}and Yehuda Shalom

^{[2]}(1) University of California, San Diego, United States

(2) Tel Aviv University, Israel

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $\Gamma$ and $\Lambda$ 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 stabilizer of any non-zero point in the Lie algebra of $G$ under the adjoint action of $G$ is amenable (e.g., if $G=\operatorname{SL}_2(\mathbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $\Gamma\curvearrowright G/\Lambda$ is either hyperfinite or rigid.

*Keywords: *Relative property (T), homogenous spaces, II_{1} factors, equivalence relations

Ioana Adrian, Shalom Yehuda: Rigidity for equivalence relations on homogeneous spaces. *Groups Geom. Dyn.* 7 (2013), 403-417. doi: 10.4171/GGD/187