Cocycle superrigidity for profinite actions of property (T) Groups

Consider a free ergodic measure-preserving profinite action Γ↷X (i.e., an inverse limit of actions Γ↷Xn, with Xn finite) of a countable property (T) group Γ (more generally, of a group Γ which admits an infinite normal subgroup Γ0 such that the inclusion Γ0⊂Γ has relative property (T) and Γ/Γ0 is finitely generated) on a standard probability space X. We prove that if w:Γ×X→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×X→Γ×Xn, for some n. As a corollary, we show that any orbit equivalence of Γ↷X with any free ergodic measure-preserving action Λ↷Y comes from a (virtual) conjugacy of actions.