Cocycle superrigidity for coinduced actions

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 $\mathcal V.$