Outer automorphism groups of some ergodic equivalence relations

TL;DR

This paper investigates the structure of outer automorphism groups of certain ergodic equivalence relations generated by higher rank lattice actions, using advanced rigidity and measure equivalence techniques.

Contribution

It provides conditions for finiteness and triviality of Out(R) and explicitly computes this group for standard actions of higher rank lattices.

Findings

Out(R) can be finite or trivial under certain conditions

Explicit computation of Out(R) for standard lattice actions

Uses Zimmer's superrigidity, Ratner's theorem, and Gromov's measure equivalence

Abstract

Let R a be countable ergodic equivalence relation of type II_1 on a standard probability space (X,m). 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 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.