# Coarse groups, and the isomorphism problem for oligomorphic groups

## TL;DR

This paper introduces coarse groups as a new tool to analyze the complexity of the isomorphism relation among closed subgroups of the permutation group of natural numbers, revealing that oligomorphic groups have a Borel reducible isomorphism relation.

## Contribution

It develops the theory of coarse groups, establishes a duality with certain topological groups, and applies this framework to classify the isomorphism complexity of oligomorphic groups.

## Key findings

- The isomorphism relation for oligomorphic groups is Borel reducible to a countable-class equivalence relation.
- A Borel reconstruction of groups from their coarse groups is possible.
- The framework extends to a broader class of closed subgroups topologically isomorphic to oligomorphic groups.

## Abstract

Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed subgroups of $S_\infty$. We axiomatise them abstractly as structures with a ternary relation. For appropriate classes of groups, including the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular we can recover an isomorphic copy of~$G$ from $\mathcal M(G)$ in a Borel fashion. A closed subgroup $G$ of $S_\infty$ is called oligomorphic if for each $n$, its natural action on $n$-tuples of natural numbers has only finitely many orbits. We use the concept of a coarse group to show that the isomorphism relation for oligomorphic subgroups of $S_\infty$ is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of $S_\infty$ that are topologically isomorphic to oligomorphic groups.