Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal $2$-extensions of Pythagorean fields and Right Angled Artin Groups

Oussama Hamza, Christian Maire, Ján Mináč, Nguyen Duy Tân

TL;DR

The paper introduces Δ-Right Angled Artin groups (Δ-RAAGs) as a framework to model maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type (RPF fields). It proves that these maximal pro-$2$ Galois quotients are Δ-RAAGs uniquely determined by an underlying graph, and conversely that a Δ-RAAG arises from an RPF field exactly when its graph data matches, yielding a minimal, quadratic presentation for these Galois quotients. It develops the Gocha series, Zassenhaus filtrations, and cohomological properties (Koszulity, Massey vanishing, Kernel Unipotent) within the Δ-RAAG framework, and provides an explicit example of a pro-$2$ group with these properties that is not a maximal Galois quotient, illustrating the limits of the characterization. The results connect graph-theoretic data to field arithmetic, offering new realizations and obstructions for absolute Galois groups in the pro-$2$ setting with concrete cohomological and filtration descriptions.

Abstract

In this paper, we describe minimal presentations of maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type. For this purpose, we introduce a new class of pro-$2$ groups: $Δ$-Right Angled Artin groups. We show that maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type are $Δ$-Right Angled Artin groups. Conversely, let us assume that a maximal pro-$2$ quotient of an absolute Galois group is a $Δ$-Right Angled Artin group. We then show that the underlying field must be Pythagorean, formally real and of finite type. As an application, we provide an example of a pro-$2$ group which is not a maximal pro-$2$ quotient of an absolute Galois group, although it has Koszul cohomology and satisfies both the Kernel Unipotent and the strong Massey Vanishing properties. We combine tools from group theory, filtrations and associated Lie algebras, profinite version of the Kurosh Theorem on subgroups of free products of groups, as well as several new techniques developed in this work.

Maximal $2$-extensions of Pythagorean fields and Right Angled Artin Groups

TL;DR

The paper introduces Δ-Right Angled Artin groups (Δ-RAAGs) as a framework to model maximal pro- quotients of absolute Galois groups of formally real Pythagorean fields of finite type (RPF fields). It proves that these maximal pro- Galois quotients are Δ-RAAGs uniquely determined by an underlying graph, and conversely that a Δ-RAAG arises from an RPF field exactly when its graph data matches, yielding a minimal, quadratic presentation for these Galois quotients. It develops the Gocha series, Zassenhaus filtrations, and cohomological properties (Koszulity, Massey vanishing, Kernel Unipotent) within the Δ-RAAG framework, and provides an explicit example of a pro- group with these properties that is not a maximal Galois quotient, illustrating the limits of the characterization. The results connect graph-theoretic data to field arithmetic, offering new realizations and obstructions for absolute Galois groups in the pro- setting with concrete cohomological and filtration descriptions.

Abstract

In this paper, we describe minimal presentations of maximal pro- quotients of absolute Galois groups of formally real Pythagorean fields of finite type. For this purpose, we introduce a new class of pro- groups: -Right Angled Artin groups. We show that maximal pro- quotients of absolute Galois groups of formally real Pythagorean fields of finite type are -Right Angled Artin groups. Conversely, let us assume that a maximal pro- quotient of an absolute Galois group is a -Right Angled Artin group. We then show that the underlying field must be Pythagorean, formally real and of finite type. As an application, we provide an example of a pro- group which is not a maximal pro- quotient of an absolute Galois group, although it has Koszul cohomology and satisfies both the Kernel Unipotent and the strong Massey Vanishing properties. We combine tools from group theory, filtrations and associated Lie algebras, profinite version of the Kurosh Theorem on subgroups of free products of groups, as well as several new techniques developed in this work.

Paper Structure

This paper contains 21 sections, 3 theorems, 89 equations.

Table of Contents

  1. Introductory results on $\Delta$-RAAGs
  2. Filtrations on $\Delta$-RAAGs
  3. Third Zassenhaus quotient of $\Delta$-RAAGs
  4. Gocha series and associated graded algebras
  5. Example: $\delta_{{\mathbf{z}}_0}$-action
  6. Pythagorean fields
  7. Semi-direct product and the class ${\mathcal{P}}$
  8. Structural results
  9. Structure of $G_L$
  10. Semi-direct products
  11. Coproducts
  12. Conclusion
  13. $\Delta$-RAAG theory and Pythagorean fields
  14. Structure of $Gr({\mathcal{P}})$
  15. Koszulity, cohomology and gocha series for groups in ${\mathcal{P}}$
  16. ...and 6 more sections

Key Result

Theorem A

If $K$ is a RPF field, then the exact sequence split Pyt splits. The pro-$2$-group $G_L$ is RAAG, and $G_K$ is $\Delta$-RAAG. Conversely, let $K$ be a field, and assume that $G_K$ is a $\Delta$-RAAG. Then $K$ is a RPF field. Moreover, $G_K$ is uniquely determined by its underlying graph.

Theorems & Definitions (36)

  • Theorem A: Theorems \ref{['mainpyt']} and \ref{['Graph and Pyt']}
  • Corollary : Proposition \ref{['koszulpyt']}
  • Theorem B: Theorem \ref{['counterexample']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 26 more