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

Abstract

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