André Nies: Topological isomorphism for classes of closed subgroups of the group of permutations of N

Tuesday, November 20, 2018, 10:30
Seminar room N0.003, Mathematical Institute, University of Bonn

Speaker: André Nies (Auckland)

Title: Topological isomorphism for classes of closed subgroups of the group of permutations of N

Abstract:

The closed subgroups of the group of permutations of $\mathbb{N}$ coincide with the automorphism groups of structures with domain $\mathbb{N}$. We consider natural Borel classes of such groups, such as being profinite (each orbit is finite), or being oligomorphic (for each $k$ there are only finitely many $k$-orbits). We work towards classifying the complexity of their isomorphism relation in the sense of Borel reducibility.

For each class, work with A. Kechris and K. Tent (J. Symb. Logic, in press) shows that the topological isomorphism relation is Borel below the isomorphism relation among countable graphs. For the class of profinite groups, we show that this bound is sharp. On the other hand, for oligomorphic groups, work with Schlicht and Tent from this year shows that the isomorphism relation is below a Borel equivalence relation with only countable classes, and hence much lower than graph isomorphism. A lower bound, other than the identity relation on the set of reals, remains unknown.

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