# 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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