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On the superrigidity of malleable actions with spectral gap. (English) Zbl 1222.46048
Summary: We prove that, if a countable group $$\Gamma$$ contains infinite commuting subgroups $$H, H^{\prime}\subset \Gamma$$ with $$H$$ non-amenable and $$H^{\prime}$$ “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, $$H^{\prime}$$ 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.

##### MSC:
 46L36 Classification of factors 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 28D15 General groups of measure-preserving transformations 46L55 Noncommutative dynamical systems
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