Cocycle and Orbit Equivalence Superrigidity for Malleable Actions of w-Rigid Groups

Abstract

We prove that if a countable discrete group $Γ$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $Γ= H \times H'$ with $H$ an infinite Kazhdan group and $H'$ arbitrary), and $\Cal V$ is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. $\Cal V$ countable discrete, or separable compact), then any $\Cal V$-valued measurable cocycle for a measure preserving action $Γ\curvearrowright X$ of $Γ$ on a probability space $(X,μ)$ which is weak mixing on $H$ and {\it s-malleable} (e.g. the Bernoulli action $Γ\curvearrowright [0,1]^Γ$) is cohomologous to a group morphism of $Γ$ into $\Cal V$. We use the case $\Cal V$ discrete of this result to prove that if in addition $Γ$ has no non-trivial finite normal subgroups then any orbit equivalence between $Γ\curvearrowright X$ and a free ergodic measure preserving action of a countable group $Λ$ is implemented by a conjugacy of the actions, with respect to some group isomorphism $Γ\simeq Λ$.