Cocycle superrigidity for translation actions of product groups

Abstract

Let G be either a profinite or a connected compact group, and Γ, Λ be finitely generated dense subgroups. Assuming that the left translation action of Γ on G is strongly ergodic, we prove that any cocycle for the left-right translation action of Γ × Λ 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 Λ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE-and W *-superrigidity. In particular, we obtain the first examples of compact actions of F2 × F2 which are W *-superrigid.