Outer automorphism groups of some equivalence relations

Abstract

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