Quasinormal subrelations of ergodic equivalence relations

Danilenko, Alexandre

doi:10.1090/S0002-9939-98-04909-0

Quasinormal subrelations of ergodic equivalence relations
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by Alexandre I. Danilenko PDF

Proc. Amer. Math. Soc. 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.

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