Isomorphism of locally compact Polish metric structures

TL;DR

This paper investigates the complexity of isomorphism relations in locally compact Polish metric structures, showing they are classifiable by countable structures and relating isometry to graph isomorphism, with implications for Borel reducibility.

Contribution

It proves that isomorphism on locally compact Polish metric structures is always classifiable by countable structures, linking it to graph isomorphism and analyzing Borel reducibility.

Findings

Isomorphism on such classes is classifiable by countable structures.

Isometry of locally compact Polish metric spaces reduces to graph isomorphism.

Borel reducibility of certain isomorphism relations to equality on hereditarily countable sets.

Abstract

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\pmb{\Pi}^0_{\alpha+1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha$, $\alpha \geq 2$. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures.