Rigidity for equivalence relations on homogeneous spaces

Abstract

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices F and Λ in a semisimple Lie group G with finite center and no compact factors we prove that the action F G/Λ is rigid. If in addition G has property (T) then we derive that the von Neumann algebra L∞ G/Λ 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 D SL2(R), then any ergodic subequivalence relation of the orbit equivalence relation of the action F G/Λ is either hyperfinite or rigid. © European Mathematical Society.