On the superrigidity of malleable actions with spectral gap

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.