On isometry and isometric embeddability between ultrametric Polish spaces

TL;DR

This paper investigates the complexity of isometry and isometric embeddability relations among ultrametric Polish spaces with fixed sets of distances, revealing their dependence on the order type of the distance set and solving open problems.

Contribution

It characterizes the Borel reducibility complexity of isometry and embeddability relations based on the order type of the distance set, including resolving a long-standing open problem.

Findings

Isometry is Borel bireducible with countable graph isomorphism when D contains a decreasing sequence.

Isometric embeddability has maximal complexity among analytic quasi-orders for certain D.

The complexity of these relations varies with the order type of D, forming sequences of Borel equivalence relations and quasi-orders.

Abstract

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length $\omega_1$ which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least $\omega+3$. We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.