Outer automorphism groups of some ergodic equivalence relations

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.

41 pages

References in corpus (1)

• Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

Cited by in corpus (11)

• A survey of Measured Group Theory
• Relative Property (T) Actions and Trivial Outer Automorphism Groups
• Product groups acting on manifolds
• Borel reductions of profinite actions of SL(n,Z)
• On Popa's Cocycle Superrigidity Theorem
• Cocycle and Orbit Equivalence Superrigidity for Malleable Actions of w-Rigid Groups
• Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups
• Cocycle Superrigidity for Profinite Actions of Property (T) Groups
• L2-torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence
• Orbit equivalence rigidity and bounded cohomology
• When is a group action determined by its orbit structure?