Rigidity for equivalence relations on homogeneous spaces

We study Popa’s notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $\mathrm{\Gamma },\mathrm{\Lambda }$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $\mathrm{\Gamma }\curvearrowright G\mathrm{/}\mathrm{\Lambda }$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\mathrm{\infty }}(G\mathrm{/}\mathrm{\Lambda })⋊\mathrm{\Gamma }$ 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 $G=S{L}_2(\mathbb{R})$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $\mathrm{\Gamma }\curvearrowright G\mathrm{/}\mathrm{\Lambda }$ is either hyperfinite or rigid.