  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.
