Cocycle superrigidity for translation actions of product groups

Abstract

Let $G$ be either a profinite or a connected compact group, and $\Gamma ,\Lambda$ be finitely generated dense subgroups. Assuming that the left translation action of $\Gamma$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $\Gamma \times \Lambda$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $\Lambda$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W${}^{\ast }$- superrigidity. In particular, we obtain the first examples of compact actions of ${\mathbb{F}}_2\times {\mathbb{F}}_2$ which are W${}^{\ast }$-superrigid.