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Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance

Ondřej Kurka

TL;DR

This work analyzes Borel reducibility of pseudometrics on standard Borel spaces via orbit pseudometrics, linking them to the Gromov-Hausdorff distance. It proves a universality result: the Gromov-Hausdorff distance $\varrho_{GH}$ is Borel-u.c. bireducible with a universal orbit pseudometric, achieved through a construction that codes orbit-pseudometric information into a Polish-space coding $F(\mathbb{U})$ and relates it to $\varrho_{GH}$. The core contribution, Theorem $\mathrm{thmmain3}$, constructs a Borel map $p\mapsto Y_p$ with $\varrho_{GH}(Y_p,Y_q)$ sandwiched between $\varrho_{G,d}(p,q)$ and twice that value, establishing a robust reduction between the orbit metric and GH distance. The results connect the GH distance with the isometry action on the Urysohn coding and extend to a Banach-space setting, while open questions address the necessity of the continuous-pseudometric system and the limits of such reductions for broader classes of groups.

Abstract

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by Cúth, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of an orbit equivalence relation. It is well known that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. We prove a version of this result for pseudometrics, showing that the Gromov-Hausdorff distance of Polish metric spaces is bireducible with a universal element in a certain class of orbit pseudometrics.

Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance

TL;DR

This work analyzes Borel reducibility of pseudometrics on standard Borel spaces via orbit pseudometrics, linking them to the Gromov-Hausdorff distance. It proves a universality result: the Gromov-Hausdorff distance is Borel-u.c. bireducible with a universal orbit pseudometric, achieved through a construction that codes orbit-pseudometric information into a Polish-space coding and relates it to . The core contribution, Theorem , constructs a Borel map with sandwiched between and twice that value, establishing a robust reduction between the orbit metric and GH distance. The results connect the GH distance with the isometry action on the Urysohn coding and extend to a Banach-space setting, while open questions address the necessity of the continuous-pseudometric system and the limits of such reductions for broader classes of groups.

Abstract

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by Cúth, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of an orbit equivalence relation. It is well known that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. We prove a version of this result for pseudometrics, showing that the Gromov-Hausdorff distance of Polish metric spaces is bireducible with a universal element in a certain class of orbit pseudometrics.
Paper Structure (5 sections, 13 theorems, 63 equations)

This paper contains 5 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The Construction
  4. Comments and questions
  5. Banach spaces setting

Key Result

Theorem 1.1

Let $G$ be a Polish group acting continuously on a Polish space $X$. Let $d$ be a lower semicontinuous pseudometric on $X$ such that $d(x, y) = d(gx, gy)$ for any $x, y \in X$ and $g \in G$. Moreover, let there be a system $\Phi$ of continuous pseudometrics on $X$ such that $d = \sup_{\varphi \in \P Then $\varrho_{G, d}$ is Borel-u.c. reducible to the Gromov-Hausdorff distance $\varrho_{GH}$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1: Katětov katetov
  • Lemma 2.2
  • proof
  • Theorem 2.3: cdk1
  • Theorem 2.4: Melleray melleray
  • Theorem 3.1
  • Claim 3.2
  • ...and 29 more