Outer automorphism groups of some ergodic equivalence relations

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 illustrate the results on the standard examples. Restrictions of such relations give an uncountable family of ergodic II_1-relations without outer automorphisms. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction.

Authors: A. Furman,Bibliographical: Comment. Math. Helv.

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[Papers] [Alex Furman]