Cocycle superrigidity for profinite actions of irreducible lattices

TL;DR

This paper establishes conditions under which certain actions of irreducible lattices in product groups are virtually cocycle superrigid, implying strong rigidity properties for these actions and their associated von Neumann algebras.

Contribution

It provides new necessary conditions for cocycle superrigidity of profinite actions of irreducible lattices, extending rigidity results to broader classes of groups and actions.

Findings

Proves virtual cocycle superrigidity for specific lattice actions

Shows ergodic profinite actions of SL_2(Z[S^{-1}]) are superrigid

Establishes implications for W*-superrigidity of associated von Neumann algebras

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 $w:\Gamma\times K\rightarrow\Delta$ 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 $\Gamma=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually cocycle superrigid and virtually W$^*$-superrigid, for any finite nonempty set of primes $S$.