## Quasinormal subrelations of ergodic equivalence relations

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- by Alexandre I. Danilenko PDF
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**126**(1998), 3361-3370 Request permission

## Abstract:

We introduce a notion of quasinormality for a nested pair $\mathcal {S}\subset \mathcal {R}$ of ergodic discrete hyperfinite equivalence relations of type $II_{1}$. (This is a natural extension of the normality concept due to Feldman-Sutherland-Zimmer.) Such pairs are characterized by an irreducible pair $F\subset Q$ of countable amenable groups or rather (some special) their Polish closure $\overline {F}\subset \overline {Q}$. We show that “most” of the ergodic subrelations of $\mathcal {R}$ are quasinormal and classify them. An example of a nonquasinormal subrelation is given. We prove as an auxiliary statement that two cocycles of $\mathcal {R}$ with dense ranges in a Polish group are weakly equivalent.## References

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## Additional Information

**Alexandre I. Danilenko**- Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom square 4, Kharkov, 310077, Ukraine
- MR Author ID: 265198
- Email: danilenko@ilt.kharkov.ua
- Received by editor(s): April 10, 1997
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 3361-3370 - MSC (1991): Primary 28D99, 46L55
- DOI: https://doi.org/10.1090/S0002-9939-98-04909-0
- MathSciNet review: 1610944