Coarse groups, and the isomorphism problem for oligomorphic groups

Let S-infinity denote the topological group of permutations of the natural numbers. A closed subgroup G of S-infinity is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of S-infinity in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup G of S-infinity the coarse group M(G) is the structure with domain the cosets of open subgroups of G, and a ternary relation AB subset of C. This structure derived from G was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, T. Symbolic Logic 83(3) (2018) 1190-1203, Sec. 3.3]. 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-infinity. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also 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 its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of S-infinity 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-infinity that are topologically isomorphic to oligomorphic groups.