Finite central extensions of o-minimal groups

Abstract

We answer in the affirmative a conjecture of Berarducci et al. (Confl. Math. 2(4): 473–496, 2010) for solvable groups, which is an o-minimal version of a particular case of Milnor’s isomorphism conjecture (Milnor, Comment Math Helv 58(1): 72–85, 1983). We prove that every abstract finite central extension of a definably connected solvable definable group in an o-minimal structure is equivalent to a definable (hence topological) finite central extension. The proof relies on an o-minimal adaptation of the higher inflation-restriction exact sequence due to Hochschild and Serre. As in Milnor (Comment Math Helv 58(1): 72–85, 1983), we also prove in o-minimal expansions of real closed fields that the conjecture reduces to definably simple groups.