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Proceedings of the American Mathematical Society

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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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Alexandre I. Danilenko
Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom square 4, Kharkov, 310077, Ukraine
MR Author ID: 265198

Received by editor(s): April 10, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society