Outer automorphism groups of some equivalence relations

Alex Furman
2005 Commentarii Mathematici Helvetici  
Let R a be countable ergodic equivalence relation of type II 1 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
more » ... ard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction. (2000) . 37A20, 28D15, 22E40, 22F50, 46L40. Mathematics Subject Classification Keywords. Ergodic equivalence relations, higher rank Lie group, lattices, outer automorphisms. * Supported in part by NSF grants DMS-0049069 and DMS-0094245. Elements of Out R represent measurable ways to permute R-classes on (X, µ). The full group Inn R is always very large (see Lemma 2.1). For the unique amenable equivalence relation R am of type II 1 the outer automorphism group Out R am is also enormous. The purpose of this paper is to analyze Out R X, for orbit relations R X, generated by m.p. ergodic actions of higher rank lattices, in particular presenting many natural examples of relations R with trivial Out R. Such examples were first constructed by S. Gefter in [6], [7] (Theorem 1.5 below). Prior to stating the results let us define two special subgroups in Out R, in the case where R is the orbit relation R X, generated by some measure-preserving action (X, µ, ) of some countable group . In such a situation consider the group Aut(X, ) of action automorphisms of the system (X, µ, )
doi:10.4171/cmh/10 fatcat:smep346cefe4neoumpbymzf5xa