Cocycle superrigidity for translation actions of product groups

TL;DR

This paper establishes cocycle superrigidity results for translation actions of product groups on various types of groups, leading to new examples of superrigid actions with applications in orbit equivalence and operator algebras.

Contribution

It proves cocycle superrigidity for translation actions of product groups on profinite, compact, and simple Lie groups, extending previous results and providing new superrigidity examples.

Findings

Cocycle superrigidity holds for actions on profinite and compact groups.

First examples of W*-superrigid actions of _2 d7 _2.

Applications to orbit equivalence and von Neumann algebra superrigidity.

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$^*$- superrigidity. In particular, we obtain the first examples of compact actions of $\mathbb F_2\times\mathbb F_2$ which are W$^*$-superrigid.