Outer automorphism groups of some equivalence relations

Abstract

Let $\Rel$ a be countable ergodic equivalence relation of type ${\rm II}_1$ on a standard probability space $(X,\mu)$. The group $\Rout\Rel$ of \emph{outer automorphisms} of $\Rel$ consists of all invertible Borel measure preserving maps of the space which map $\Rel$-classes to $\Rel$-classes modulo those which preserve almost every $\Rel$-class. We analyze the group $\Rout\Rel$ for relations $\Rel$ generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of $\Rout\Rel$ and explicitly computing $\Rout\Rel$ for the standard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction.