On subrelations of ergodic measured type III equivalence relations

Abstract

We discuss the classification up to orbit equivalence of inclusions S ⊂ R of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of S ⊂ R), an ergodic nonsingular R-flow V and a homomorphism of G to the centralizer of V . 0. Introduction. We consider nonsingular discrete ergodic hyperfinite equivalence relations on a standard measure space. Our concern is to classify pairs (R,S) where R is an ergodic equivalence relation and S ⊂ R is a subrelation of finite index (which means that the R-equivalence class of a.e. point consists of finitely many S-classes), up to orbit equivalence. This problem is closely related to the classification of subfactors in von Neumann algebras theory. For a single equivalence relation R the problem was solved by H. Dye [Dy] and W. Krieger [Kr] in terms of the associated flows. Then, in the case where R is of type II1, J. Feldman, C. Sutherland, and R. Zimmer [FSZ] provided a simple classification of ergodic R-subrelations of finite index and normal R-subrelations of arbitrary index. (We remark that in an earlier paper [Ge] M. Gerber classified R-subrelations of finite index in a different—but equivalent—context of finite extensions of ergodic probability preserving transformations.) These results were further extended in [Da1, §4] and [Da2], where quasinormal subrelations of type II1 were introduced and studied. Recently, T. Hamachi considered finite index subrelations of a type III0 equivalence relation R, introduced a system of invariants for orbit equiva2000 Mathematics Subject Classification: 28D99, 46L55. The research was supported in part by INTAS 97-1843.