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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0894-0347-07-00578-4
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
MR Author ID: 141080
Email: popa@math.ucla.edu

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