Cocycle superrigidity for coinduced actions

TL;DR

This paper establishes a cocycle superrigidity result for a broad class of coinduced actions, demonstrating that certain cocycles are essentially homomorphisms, under conditions involving property (T) or amenability and product structures.

Contribution

It proves a new cocycle superrigidity theorem for coinduced actions using Popa's deformation/rigidity theory, extending previous results to larger classes of groups and actions.

Findings

Cocycle superrigidity holds for coinduced actions under specified conditions.

Any cocycle to a $in$ group is cohomologous to a homomorphism.

Results apply to groups with property (T) or amenable subgroups in product groups.

Abstract

We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $\Lambda$ is a subgroup of a countable group $\Gamma$, we consider a probability measure preserving action $\Lambda\curvearrowright X_0$ and let $\Gamma\curvearrowright X$ be the coinduced action. Assume either that $\Gamma$ has property (T) or that $\Lambda$ is amenable and $\Gamma$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $\Gamma\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is cohomologous to a homomorphism from $\Gamma$ to $\mathcal V.$