Rigidity for equivalence relations on homogeneous spaces

Adrian Ioana, Yehuda Shalom

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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.

Original languageEnglish
Pages (from-to)403-417
Number of pages15
JournalGroups, Geometry, and Dynamics
Volume7
Issue number2
DOIs
StatePublished - 2013
Externally publishedYes

Funding

FundersFunder number
Directorate for Mathematical and Physical Sciences0701639, 1007227

    Keywords

    • Equivalence relations.
    • Homogenous spaces
    • II factors
    • Relative property (T)

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