Outer automorphism groups of some ergodic equivalence relations

2003/12/17 by Alex Furman, Furman, Alex
Mathematics · #22E40 #22F50 #28D15 #37A20 #46L40 #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA) #math.DS #math.OA #msc:22E40 #msc:22F50 #msc:28D15 #msc:37A20 #msc:46L40

paper · pdf · doi:10.48550/arxiv.math/0312346

41 pages

arxiv created 2003/12/17 · arxiv updated 2009/12/01

Abstract

Let R a be countable ergodic equivalence relation of type II₁ 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.

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