Cocycle superrigidity for profinite actions of irreducible lattices (Q2694836)

Summary: 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 \(w : \Gamma \times K \to \Delta\) with values in a countable group \(\Delta\) is cohomologous to a cocycle which factors through the map \(\Gamma \times K \to \Gamma \times K_0\) 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 \(W^\ast\)-superrigid for any finite nonempty set of primes \(S\).