Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

We prove that if a countable discrete group Gamma is w- rigid, i. e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Gamma = SL(2, Z) x Z(2), or Gamma = H x H' with H an infinite Kazhdan group and H' arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any V- valued measurable cocycle for a measure preserving action Gamma curved right arrow X of Gamma on a probability space ( X, mu) which is weak mixing on H and s-malleable (e.g. the Bernoulli action Gamma curved right arrow [ 0, 1] G) is cohomologous to a group morphism of Gamma into V. We use the case V discrete of this result to prove that if in addition Gamma has no non- trivial finite normal subgroups then any orbit equivalence between Gamma curved right arrow X and a free ergodicmeasure preserving action of a countable group Lambda is implemented by a conjugacy of the actions, with respect to some group isomorphism Gamma similar or equal to Lambda.