JournalscmhVol. 80 , No. 1DOI 10.4171/cmh/10

Outer automorphism groups of some equivalence relations

  • Alex Furman

    University of Illinois at Chicago, USA
Outer automorphism groups of some equivalence relations cover

Abstract

Let \Rel\Rel a be countable ergodic equivalence relation of type II1{\rm II}_1 on a standard probability space (X,μ)(X,\mu). The group \Rout\Rel\Rout\Rel of \emph{outer automorphisms} of \Rel\Rel consists of all invertible Borel measure preserving maps of the space which map \Rel\Rel-classes to \Rel\Rel-classes modulo those which preserve almost every \Rel\Rel-class. We analyze the group \Rout\Rel\Rout\Rel for relations \Rel\Rel generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of \Rout\Rel\Rout\Rel and explicitly computing \Rout\Rel\Rout\Rel for the standard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction.