On the superrigidity of malleable actions with spectral gap

Abstract

We prove that if a countable group Γ \Gamma contains infinite commuting subgroups H , H ′ ⊂ Γ H, H’\subset \Gamma with H H non-amenable and H ′ H’ “weakly normal” in Γ \Gamma , then any measure preserving Γ \Gamma -action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli Γ \Gamma -action) is cocycle superrigid. If in addition H ′ H’ can be taken non-virtually abelian and Γ ↷ X \Gamma \curvearrowright X is an arbitrary free ergodic action while Λ ↷ Y = T Λ \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 _{1} factors L ∞ X ⋊ Γ ≃ L ∞ Y ⋊ Λ L^{\infty }X \rtimes \Gamma \simeq L^{\infty }Y \rtimes \Lambda comes from a conjugacy of the actions.