Cocycle superrigidity for profinite actions of property (T) Groups

Abstract

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 $w:\Gamma \times X\to \Lambda$ 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 $\Gamma \times X\to \Gamma \times X_n$, 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.