Cocycle Superrigidity for Profinite Actions of Property (T) Groups

Abstract

Consider a free ergodic measure preserving profinite action $Γ\curvearrowright X$ (i.e. an inverse limit of actions $Γ\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $Γ$ (more generally of a group $Γ$ which admits an infinite normal subgroup $Γ_0$ such that the inclusion $Γ_0\subsetΓ$ has relative property (T) and $Γ/Γ_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:Γ\times X\to Λ$ is a measurable cocycle with values in a countable group $Λ$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $Γ\times X\to Γ\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $Γ\curvearrowright X$ with any free ergodic measure preserving action $Λ\curvearrowright Y$ comes from a (virtual) conjugacy of actions.