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<p>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.</p>

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Finite central extensions of o-minimal groups

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  • 发表日期：

    2025年6月10日

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  • Elías Baro,
  • Daniel Palacín

摘要

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.