Outer automorphism groups of some ergodic equivalence relations

Let R a be countable ergodic equivalence relation of type II_1 on a standard probability space (X,m). The group Out® of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map R-classes to R-classes modulo those which preserve almost every R-class. We analyze the group Out® for relations R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of Out® and explicitly computing this group for the standard actions. The method is based on Zimmer’s superrigidity for measurable cocycles, Ratner’s theorem and Gromov’s Measure Equivalence construction.