Rigidity for equivalence relations on homogeneous spaces

2010/10/19 by Adrian Ioana, Yehuda Shalom, Ioana, Adrian +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #math.DS #math.GR #math.OA

paper · pdf · doi:10.48550/arxiv.1010.3778

arxiv created 2010/11/03 · arxiv updated 2010/11/05

Abstract

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

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