Finite central extensions of o-minimal groups

TL;DR

This paper proves that in o-minimal structures, finite central extensions of definably connected solvable groups are definably equivalent to definable extensions, confirming a conjecture for this class of groups.

Contribution

It establishes that all finite central extensions of solvable definable groups in o-minimal structures are definably equivalent to definable extensions, confirming a conjecture for solvable groups.

Findings

Finite central extensions of solvable groups are definably equivalent to definable extensions.

The proof uses an o-minimal adaptation of the Hochschild-Serre inflation-restriction sequence.

The conjecture reduces to the case of definably simple groups in o-minimal expansions of real closed fields.

Abstract

We answer in the affirmative a conjecture of Berarducci, Peterzil and Pillay \cite{BPP10} for solvable groups, which is an o-minimal version of a particular case of Milnor's isomorphism conjecture \cite{jM83}. 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 \cite{jM83}, we also prove in o-minimal expansions of real closed fields that the conjecture reduces to definably simple groups.