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Complexity questions for classes of closed subgroups of the group of permutations of N Nies, Andre


The closed subgroups of the group of permutations of N coincide with the automorphism groups of countable structures. We consider classes of such groups, such as being oligomorphic, or being topologically finitely generated. We study their complexity in the sense of descriptive set theory. If the class is Borel, we next consider the complexity of the topological isomorphism relation. For instance, the classes of locally compact and of oligomorphic groups are both Borel. In either case, the isomorphism relation is bounded in complexity by the problem of deciding whether two countable graphs are isomorphic. In the first case, and even restricted to compact (i.e. profinite separable) groups, this upper bound is known to be sharp. The upper bounds have been obtained independently by Rosendal and Zielinski (arXiv, Oct. 2016). This is joint work with A. Kechris and K. Tent.

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