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April 16, 2026Orbita Mathematicae0 citationsOpen Access

On dense orbits in the space of subequivalence relations

Key Points

To characterize which subequivalence relations have dense orbits in the space of subequivalence relations of ergodic hyperfinite p.m.p. equivalence relations.
Endowed the space of subequivalence relations with Polish topology.
Focused on p.m.p. equivalence relations and their natural actions.
Characterized subequivalence relations with dense orbits under full group and automorphism group actions.
Performed Borel complexity calculations using the uniform metric.
Identified specific subequivalence relations exhibiting dense orbits.
Established that all orbits under the full group action are meager.
Conducted Borel complexity analyses that respond to previous inquiries in the field.

Abstract

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.

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