Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 3-34.

Canonical subrelations of ergodic equivalence relations-subrelations

Authors: Toshihiro Hamachi
Author institution: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Chuo-ku, Fukuoka 810--8560, Japan

Summary: Given an ergodic measured discrete equivalence relation

${\cal R}$ and an ergodic subrelation

${\cal S} \subset {\cal R}$ of finite index, C. Sutherland showed that they are represented by the cross products

${\cal P}\Rtimes _{\alpha }G$ and

${\cal P} \Rtimes _{\alpha }H$ of an ergodic subrelation

${\cal P} \subset {\cal S}$ by a finite group outer action

$\alpha_{G}$ and a subgroup action

$\alpha_{H}$ . This result is strengthe ned in the sense that the subgroup

$H$ may be chosen so that it does not contain any non-trivial normal subgroup of

$G$ and that the collection

$\{{\cal P}, H \subset G, \alpha_{G} \}$ is invariant for the orbit equivalence of the pair o f

${\cal R}$ and

${\cal S}$ . In amenable case of type II

$_{1}$ , a complete invariant for the orbit equivalence o f pairs of an ergodic measured discrete equivalence relation and an ergodic subrelation of finite index is obta ined.

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