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On the superrigidity of malleable actions with spectral gap

Author: Sorin Popa
Journal: J. Amer. Math. Soc. 21 (2008), 981-1000
MSC (2000): Primary 46L35; Secondary 37A20, 22D25, 28D15
Published electronically: September 26, 2007
MathSciNet review: 2425177
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Abstract: We prove that if a countable group $ \Gamma $ contains infinite commuting subgroups $ H, H'\subset \Gamma $ with $ H$ non-amenable and $ H'$ ``weakly normal'' in $ \Gamma $, then any measure preserving $ \Gamma $-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli $ \Gamma $-action) is cocycle superrigid. If in addition $ H'$ can be taken non-virtually abelian and $ \Gamma \curvearrowright X$ is an arbitrary free ergodic action while $ \Lambda \curvearrowright Y=\mathbb{T}^{\Lambda }$ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II$ _{1}$ factors $ L^{\infty }X \rtimes \Gamma \simeq L^{\infty }Y \rtimes \Lambda $ comes from a conjugacy of the actions.

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Additional Information

Sorin Popa
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-155505

Keywords: von Neumann algebras, II$_{1}$ factors, Bernoulli actions, spectral gap, orbit equivalence, cocycles
Received by editor(s): October 24, 2006
Published electronically: September 26, 2007
Additional Notes: Research was supported in part by NSF Grant 0601082.
Article copyright: © Copyright 2007 American Mathematical Society

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