Quasinormal subrelations of ergodic equivalence relations

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.