October 30
Russell Miller,
CUNY
Computable categoricity for profinite groups
Infinite profinite groups $G$ have size continuum and therefore do not admit examination under the usual notions of computable structure theory. However, they are naturally presented by finite-branching trees, with group elements represented by infinite paths through the tree. For such presentations, we say that $G$ is computably categorical if, for every two computable tree presentations $T_0$ and $T_1$ of $G$ (with computable finite branching and no terminal nodes), there is a Turing functional $\Phi$ that maps paths $Q$ through $T_0$ to paths $\Phi^Q$ through $T_1$ so that the map $Q\mapsto\Phi^Q$ is an isomorphism of groups. In short, all computable tree presentations of $G$ are isomorphic via type-2-computable functions. We will begin with results, joint with Jason Block, that explain how we came to study this topic. Then we will connect computable categoricity for profinite groups with computable categoricity for countable structures, using a scheme for presenting inverse systems of profinite groups that was devised by Cherlin, van den Dries, and Macintyre. This will yield various results about categoricity for the tree presentations.