Logic Seminar

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a system of finite groups carrying the discrete topology. An example is the additive group of the 2-adic integers. We study profinite groups from a logician's point of view. Lubotzky and Jarden showed that topologically f.g. groups are given by their first-order theory. We raise the question whether a single sentence can suffice. Next, in the setting of computable profinite groups, the Haar measure is computable, so the usual notions of algorithmic randomness can be defined. We consider the strength of randomness necessary for effective versions of "almost everywhere" type theorems to hold (with Fouche). Finally, we consider the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs. The latter result can be adapted to locally compact closed subgroups of the group of permutations on N (joint with Kechris and Tent). This result has been obtained independently and by different means in work of Rosendal and Zielinski (arXiv 1610.00370, Oct 2016).