Cocycle Superrigidity for Profinite Actions of Property (T) Groups

Consider a free ergodic measure preserving profinite action $\mathrm{\Gamma }\curvearrowright X$ (i.e. an inverse limit of actions $\mathrm{\Gamma }\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\mathrm{\Gamma }$ (more generally of a group $\mathrm{\Gamma }$ which admits an infinite normal subgroup ${\mathrm{\Gamma }}_0$ such that the inclusion ${\mathrm{\Gamma }}_0\subset \mathrm{\Gamma }$ has relative property (T) and $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\mathrm{\Gamma }\times X\to \mathrm{\Lambda }$ is a measurable cocycle with values in a countable group $\mathrm{\Lambda }$, then $w$ is cohomologous to a cocycle $w^{\mathrm{\prime }}$ which factors through the map $\Gamma\times X\to \Gamma\times X_n$,forsome$n$. As a corollary, we show that any orbit equivalence of $\mathrm{\Gamma }\curvearrowright X$ with any free ergodic measure preserving action $\mathrm{\Lambda }\curvearrowright Y$ comes from a (virtual) conjugacy of actions.