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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Quasinormal subrelations of ergodic
equivalence relations


Author: Alexandre I. Danilenko
Journal: Proc. Amer. Math. Soc. 126 (1998), 3361-3370
MSC (1991): Primary 28D99, 46L55
MathSciNet review: 1610944
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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.


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Additional Information

Alexandre I. Danilenko
Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom square 4, Kharkov, 310077, Ukraine
Email: danilenko@ilt.kharkov.ua

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04909-0
PII: S 0002-9939(98)04909-0
Received by editor(s): April 10, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society