Oligomorphic groups are essentially countable

We study the complexity of the isomorphism relation on classes of closed subgroups of $S_\infty$, the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. A closed subgroup $G$ of $S_\infty$ is called $\mathit{oligomorphic}$ if for each $n$, its natural action on $n$-tuples of natural numbers has only finitely many orbits. We 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 groups topologically isomorphic to oligomorphic subgroups of $S_\infty$. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal{M}(G)$ is the structure whose domain consists of cosets of some open subgroups of $G$ and a single ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, this translates $G$ into a countable coarse group structure $\mathcal{M}(G)$ coding $G$. Coarse groups form our main tool in studying closed subgroups of $S_\infty$ with only countably many open subgroups.