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Finite central extensions of o-minimal groups

Elías Baro, Daniel Palacín

TL;DR

The paper proves Berarducci–Peterzil–Pillay's conjecture for solvable definable groups in o-minimal structures, showing that every abstract finite central extension of a definably connected solvable group is naturally interpretable in the ambient structure. It introduces a definable second cohomology $H^2_d(G,Z)$ and establishes a definable Hochschild–Serre inflation-restriction framework, yielding an isomorphism $H^2_d(G,Z) rom H^2(G,Z)$ for solvable $G$ and a concrete description $H^2(G,Z)rac{Z^{ ext{dim}(G/N(G))}}{}$; in expansions of real closed fields, $H^2_d(G,Z)$ is finite and the theory reduces to definably simple factors. The work also provides a reduction principle: if the conjecture holds for definably simple groups, it holds for all definably connected groups in the o-minimal setting, thereby linking the solvable case to the simple-case and connecting o-minimal model theory with classical Lie-group cohomology. Collectively, these results extend Milnor-type isomorphism phenomena to the o-minimal context and offer a structured path to interpretability for finite central extensions of definable groups.

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.

Finite central extensions of o-minimal groups

TL;DR

The paper proves Berarducci–Peterzil–Pillay's conjecture for solvable definable groups in o-minimal structures, showing that every abstract finite central extension of a definably connected solvable group is naturally interpretable in the ambient structure. It introduces a definable second cohomology and establishes a definable Hochschild–Serre inflation-restriction framework, yielding an isomorphism for solvable and a concrete description ; in expansions of real closed fields, is finite and the theory reduces to definably simple factors. The work also provides a reduction principle: if the conjecture holds for definably simple groups, it holds for all definably connected groups in the o-minimal setting, thereby linking the solvable case to the simple-case and connecting o-minimal model theory with classical Lie-group cohomology. Collectively, these results extend Milnor-type isomorphism phenomena to the o-minimal context and offer a structured path to interpretability for finite central extensions of definable groups.

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.
Paper Structure (8 sections, 22 theorems, 61 equations)

This paper contains 8 sections, 22 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Generalities on o-minimal finite central extensions
  3. Natural interpretability and cocycles
  4. Definable second cohomology group
  5. A definable inflation-retraction
  6. Solvable o-minimal groups
  7. On definable central extensions in o-minimal expansions of real closed fields
  8. A reduction to simple groups

Key Result

Theorem 1

Let $G$ be a definably connected solvable definable group in an o-minimal structure $M$. Then every abstract finite central extension of $G$ is naturally interpretable in $M$.

Theorems & Definitions (54)

  • Theorem 1: Theorem \ref{['T:Sol']}
  • Conjecture
  • Theorem 2: Theorem \ref{['T:Ext-Sequence']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • proof
  • ...and 44 more