# Rigidity for equivalence relations on homogeneous spaces

### Adrian Ioana

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

Tel Aviv University, Israel

## Abstract

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.

## 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