Coarse groups, and the isomorphism problem for oligomorphic groups

Let S∞ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups S∞ in the setting of Borel reducibility between equivalence relations on Polish spaces.

Given a closed subgroup G of S∞, the coarse group M(G) is the structure with domain the cosets of open subgroups of G, and a ternary relation AB ⊆ C. If G has only countably many open subgroups, then M(G) is a countable structure. Coarse groups form our main tool in studying such closed subgroups of S∞. 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 M(G) in a Borel fashion.

A closed subgroup G of S∞ 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 isomor- phism relation for oligomorphic subgroups of S∞ 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∞ that are topologically isomorphic to oligomorphic groups.