Quasinormal subrelations of ergodic equivalence relations

Abstract

We introduce a notion of quasinormality for a nested pair S ⊂ R \mathcal {S}\subset \mathcal {R} of ergodic discrete hyperfinite equivalence relations of type I I 1 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 ⊂ Q F\subset Q of countable amenable groups or rather (some special) their Polish closure F ¯ ⊂ Q ¯ \overline {F}\subset \overline {Q} . We show that “most” of the ergodic subrelations of R \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 R \mathcal {R} with dense ranges in a Polish group are weakly equivalent.