On the superrigidity of malleable actions with spectral gap

Popa, Sorin¹

¹Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-155505

Journal of the American Mathematical Society, Tome 21 (2008) no. 4, pp. 981-1000

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-07-00578-4

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