We first explain how to endow the space of subequivalence relations of any
nonsingular countable equivalence relation with a Polish topology, extending the
framework of Kechris’ recent monograph on subequivalence relations of probability
measure-preserving (p.m.p.) countable equivalence relations. We then restrict to
p.m.p. equivalence relations and discuss dense orbits therein for the natural action of
the full group and of the automorphism group of the relation. Our main result is a
characterization of the subequivalence relations having a dense orbit in the space of
subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation.
We also show that in this setup, all orbits under the full group action are
meager. We finally provide a few Borel complexity calculations of natural
subsets in spaces of subequivalence relations using a natural metric we call the
uniform metric. This answers some questions from an earlier version of Kechris’
monograph.
