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On dense orbits in the space of subequivalence relations

François Le Maître

TL;DR

This work extends Kechris's Polish topology on the space of subequivalence relations to non-singular countable relations and specializes to p.m.p. cases to analyze dense orbits under natural actions. It provides two proofs that Sub$( ho)$ is Polish, introduces strong and lower topologies, and establishes that in the ergodic hyperfinite p.m.p. setting, a subequivalence relation has a dense full-group and automorphism orbit precisely when it is aperiodic with everywhere infinite index. It also proves meagerness of full-group orbits in Sub}_{ ext{hyp}}$( ho)$ and delivers a detailed complexity analysis using the uniform metric, showing several $F_ ho$ and $G_ ho$-hardness results and demonstrating how hyperfinite structures behave under orbit closures. The results connect orbit-density phenomena with conditional measures, index theory, and intertwining methods, and provide a framework for understanding the descriptive-set-theoretic landscape of subequivalence relations in both non-singular and probability-measure-preserving contexts.

Abstract

We first explain how to endow the space of subequivalence relations of any non-singular 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 full groups orbits 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.

On dense orbits in the space of subequivalence relations

TL;DR

This work extends Kechris's Polish topology on the space of subequivalence relations to non-singular countable relations and specializes to p.m.p. cases to analyze dense orbits under natural actions. It provides two proofs that Sub is Polish, introduces strong and lower topologies, and establishes that in the ergodic hyperfinite p.m.p. setting, a subequivalence relation has a dense full-group and automorphism orbit precisely when it is aperiodic with everywhere infinite index. It also proves meagerness of full-group orbits in Sub}_{ ext{hyp}} and delivers a detailed complexity analysis using the uniform metric, showing several and -hardness results and demonstrating how hyperfinite structures behave under orbit closures. The results connect orbit-density phenomena with conditional measures, index theory, and intertwining methods, and provide a framework for understanding the descriptive-set-theoretic landscape of subequivalence relations in both non-singular and probability-measure-preserving contexts.

Abstract

We first explain how to endow the space of subequivalence relations of any non-singular 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 full groups orbits 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.
Paper Structure (14 sections, 46 theorems, 46 equations)

This paper contains 14 sections, 46 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The Polish topology on the space of subequivalence relations in the non-singular setup
  3. The measure algebra of a standard sigma-finite space
  4. The space of subequivalence relations
  5. Strong and lower topologies
  6. Topologies on measure algebras of sigma-finite spaces
  7. Lower topologies on measure algebras
  8. Seeing that the space of subequivalence relations is closed
  9. Dense orbits for hyperfinite subequivalence relations in the p.m.p. case
  10. Preliminaries on conditional measures and index
  11. Approximating the diagonal
  12. Dense orbits in the space of hyperfinite subequivalence relations
  13. Meagerness of full group orbits
  14. The uniform metric and complexity calculations

Key Result

Theorem A

Let $\mathcal{R}$ be a non-singular equivalence relation on a standard probability space. Then its space $\mathop{\mathrm{\mathrm{Sub}}}\nolimits(\mathcal{R})$ of subequivalence relations is a Polish space for the topology induced by the measure algebra of $\mathcal{R}$.

Theorems & Definitions (117)

  • Theorem A: see Thm. \ref{['thm: sub R is Polish']}
  • Theorem B: see Cor. \ref{['cor: dense orbit in R0']}
  • Example 1.1
  • Example 1.2
  • Theorem C: see Thm. \ref{['thm: infinite index everywhere characterization']}
  • Theorem D: see Cor. \ref{['cor: orbits meager R0']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 107 more