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Subrelations of ergodic equivalence relations†

Published online by Cambridge University Press:  19 September 2008

J. Feldman ,
C. E. Sutherland  and
R. J. Zimmer
J. Feldman
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, USA
C. E. Sutherland
Affiliation:
Mathematics Department, University of New South Wales, Kensington, NSW, 2033, Australia
R. J. Zimmer
Affiliation:
Mathematics Department, University of Chicago, Chicago, Illinois 60637, USA
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Abstract

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We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 2 , June 1989 , pp. 239 - 269
Copyright
Copyright © Cambridge University Press 1989

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