Outer automorphism groups of some ergodic equivalence relations

Abstract

Let R a be countable ergodic equivalence relation of type II1 on a standard probability space (X, μ). The group Out R 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 R for relations R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of Out R and explicitly computing Out 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. © Swiss Mathematical Society.