Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

TL;DR

This paper establishes strong rigidity results for translation actions of dense subgroups on locally compact groups, showing orbit equivalence implies algebraic isomorphism and proving cocycle superrigidity under property (T).

Contribution

It proves orbit equivalence rigidity for translation actions on simply connected groups and extends results to algebraic groups and homogeneous spaces, also establishing cocycle superrigidity under property (T).

Findings

Orbit equivalence implies algebraic isomorphism of groups.

Cocycles are cohomologous to homomorphisms under property (T).

Actions are orbit equivalence superrigid, classifying all orbit equivalent actions.

Abstract

We study equivalence relations that arise from translation actions $\Gamma\curvearrowright G$ which are associated to dense embeddings $\Gamma<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $\Gamma\curvearrowright G$ is strongly ergodic, we prove that $\Gamma\curvearrowright G$ is orbit equivalent to another such translation action $\Lambda\curvearrowright H$ if and only if there exists an isomorphism $\delta:G\rightarrow H$ such that $\delta(\Gamma)=\Lambda$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of $\Gamma$ on homogeneous spaces of the form $G/\Sigma$, where $\Sigma<G$ is either a discrete or an algebraic subgroup. We also prove that if $G$ is simply connected and the action $\Gamma\curvearrowright G$ has property (T), then any cocycle $w:\Gamma\times G\rightarrow\Lambda$ with values into a countable group $\Lambda$ is cohomologous to a homomorphism $\delta:\Gamma\rightarrow\Lambda$. As a consequence, we deduce that the action $\Gamma\curvearrowright G$ is orbit equivalent superrigid: any free nonsingular action $\Lambda\curvearrowright Y$ which is orbit equivalent to $\Gamma\curvearrowright G$, is necessarily conjugate to an induction of $\Gamma\curvearrowright G$.