On isometry and isometric embeddability between ultrametric Polish spaces

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.