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Variations of Demushkin Groups that are not Absolute Galois Groups

Simone Blumer, Claudio Quadrelli

Abstract

We construct two families of examples of pro-p groups, with rather elementary presentations, that do not complete into 1-cyclotomic oriented pro-p groups. These provide brand new examples of pro-p groups that do not occur as maximal pro-p Galois groups of fields containing a root of unity of order p - and thus, as absolute Galois groups. Moreover, we show that these pro-p groups may not be ruled out as maximal pro-p Galois groups employing other cohomological properties that are known to hold for all maximal pro-p Galois groups, such as the triple Massey vanishing property, or the quadraticity of Fp-cohomology.

Variations of Demushkin Groups that are not Absolute Galois Groups

Abstract

We construct two families of examples of pro-p groups, with rather elementary presentations, that do not complete into 1-cyclotomic oriented pro-p groups. These provide brand new examples of pro-p groups that do not occur as maximal pro-p Galois groups of fields containing a root of unity of order p - and thus, as absolute Galois groups. Moreover, we show that these pro-p groups may not be ruled out as maximal pro-p Galois groups employing other cohomological properties that are known to hold for all maximal pro-p Galois groups, such as the triple Massey vanishing property, or the quadraticity of Fp-cohomology.
Paper Structure (31 sections, 26 theorems, 124 equations)

This paper contains 31 sections, 26 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Main results: new examples
  4. Galois-like properties
  5. Graded restricted Lie algebras
  6. An open question
  7. Preliminaries
  8. Notation
  9. $\mathbb{F}_p$-cohomology of pro-$p$ groups
  10. Massey products in Galois cohomology
  11. Graded group algebras and associated Lie algebras
  12. 1-cyclotomic oriented pro-$p$ groups
  13. Orientations of pro-$p$ groups
  14. Maximal pro-$p$ Galois group
  15. Detecting 1-cyclotomic oriented pro-$p$ groups
  16. ...and 16 more sections

Key Result

Theorem 1.1

Every pro-$p$ group $G$ in the family $\mathcal{F}_1$ is not 1-cyclotomic. In particular, $G$ does not occur as the maximal pro-$p$ Galois group of a field containing a root of unity of order $p$ (and also $\sqrt{-1}$ if $p=2$).

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.2
  • Remark 2.3
  • Example 2.4
  • Lemma 2.5
  • Example 2.6
  • Proposition 2.7
  • ...and 38 more