Group covers, o-minimality, and categoricity

We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case of covers of complex algebraic groups studied by Zilber and his students. We also discuss from a model-theoretic point of view the following question, related to "Milnor's conjecture": is a finite central extension (as an abstract group) of a compact Lie group also a topological extension?