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 .