2008/05/20 by Ioana, Adrian
#37A20 #46L10 #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.0805.2998
Consider a free ergodic measure preserving profinite action Γ\curvearrowright X (i.e. an inverse limit of actions Γ\curvearrowright 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\curvearrowright X with any free ergodic measure preserving action Λ\curvearrowright Y comes from a (virtual) conjugacy of actions.