Consider a free ergodic measure-preserving profinite action (i.e., an inverse limit of actions , with finite) of a countable property (T) group (more generally, of a group which admits an infinite normal subgroup such that the inclusion has relative property (T) and is finitely generated) on a standard probability space . We prove that if is a measurable cocycle with values in a countable group , then is cohomologous to a cocycle which factors through the map , for some . As a corollary, we show that any orbit equivalence of with any free ergodic measure-preserving action comes from a (virtual) conjugacy of actions.
"Cocycle superrigidity for profinite actions of property (T) Groups." Duke Math. J. 157 (2) 337 - 367, 1 April 2011. https://doi.org/10.1215/00127094-2011-008