Cocycle Superrigidity for Profinite Actions of Property (T) Groups

Consider a free ergodic measure-preserving profinite action $\Gamma\curvearrowright X$ (i.e., an inverse limit of actions $\Gamma\curvearrowright X_n$ , with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally, of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$ . We prove that if $w:\Gamma\times X\rightarrow \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$ , then $w$ is cohomologous to a cocycle $w\prime$ which factors through the map $\Gamma\times X\rightarrow \Gamma\times X_n$ , for some $n$ . As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure-preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.