Subrelations of ergodic equivalence relations

Abstract

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. © 1989, Cambridge University Press. All rights reserved.