Cocycle and Orbit Equivalence Superrigidity for Bernoulli Actions of Kazhdan Groups

We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2,Z) ⋉Z, or Γ = H × H with H an infinite Kazhdan group and H arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a Bernoulli Γ-action is cohomologous to a group morphism of Γ into V. We use this result to prove that if in addition Γ has no non-trivial finite normal subgroups, then any orbit equivalence between a Bernoulli Γ-action and a free ergodic measure preserving action of some group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ.