On the Superrigidity of Malleable Actions with Spectral Gap

Abstract

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 $L^{\infty }X⋊\Gamma \simeq L^{\infty }Y⋊\Lambda$ comes from a conjugacy of the actions.

Final version; paper appeared in Journal of the Amer. Math. Soc., 2007

References in corpus (4)

• L^2-rigidity in von Neumann algebras
• Rigidity results for wreath product II${}_1$ factors
• A 1-cohomology characterization of property (T) in von Neumann algebras
• On Popa's Cocycle Superrigidity Theorem