Cocycle superrigidity for profinite actions of irreducible lattices

Abstract

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that $\Gamma$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $\Gamma \curvearrowright K$ is “virtually” cocycle superrigid: any cocycle ${w\colon \Gamma\times K\rightarrow\Delta}$ with values in a countable group $\Delta$ is cohomologous to a cocycle which factors through the map $\Gamma\times K\rightarrow\Gamma\times K\_0$ for some finite quotient group $K\_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $\Gamma=\mathrm{SL}\_2(\mathbb Z\[S^{-1}])$ is virtually cocycle superrigid and virtually W$^\*$-superrigid for any finite nonempty set of primes $S$.