QUASI-NORMAL SUBRELATIONS OF ERGODIC EQUIVALENCE-RELATIONS

We introduce a notion of quasinormality for a nested pair S subset of R of ergodic discrete hyperfinite equivalence relations of type II1. ( This is a natural extension of the normality concept due to Feldman-Su therland-Zimmer.) Such pairs are characterized by an irreducible pair F subset of Q of countable amenable groups or rather (some special) th eir Polish closure (F) over bar subset of (Q) over bar. We show that ' 'most'' of the ergodic subrelations of 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 with dense ranges in a Polish group are weakly equivalent.