We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices and in a semisimple Lie group with finite center and no compact factors we prove that the action is rigid. If in addition has property (T) then we derive that the von Neumann algebra has property (T). We also show that if the stabilizer of any non-zero point in the Lie algebra of under the adjoint action of is amenable (e.g., if ), then any ergodic subequivalence relation of the orbit equivalence relation of the action is either hyperfinite or rigid.
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Adrian Ioana, Yehuda Shalom, Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn. 7 (2013), no. 2, pp. 403–417DOI 10.4171/GGD/187