### Oligomorphic groups are essentially countable

Logic and Set Theory Seminar

18th December 2018, 3:00 pm – 4:30 pm

Howard House, 4th Floor Seminar Room

Model theoretic properties of a countable structure are closely connected with properties of its automorphism group. For instance, the automorphism groups of ω-categorical structures on N are precisely the oligomorphic closed subgroups of Sym(N) (a permutation group is oligomorphic if for each k there are only finitely many k-orbits). In this recent project with Andre Nies and Katrin Tent, we study the complexity of topological isomorphism of oligomorphic closed subgroups of Sym(N) in the setting of Borel reducibility. Previous work of Kechris, Nies and Tent, and independently Rosendal and Zielinski, showed that this equivalence relation is below graph isomorphism. We show that it is below a Borel equivalence relation with countable equivalence classes.

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