Outer automorphism groups of some equivalence relations

Abstract

Let \( \Rel \) a be countable ergodic equivalence relation of type ${\mathrm{II}}_1$ 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.