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Presentations of Galois groups of maximal extensions with restricted ramification

Yuan Liu

TL;DR

The paper develops a comprehensive cohomological framework to analyze presentations of G_S(k), the Galois group of the maximal extension unramified outside S, across global fields and Γ-extensions. It generalizes the Euler-Poincaré characteristic to include Γ-actions and arbitrary finite simples A, introducing B_S(k,A) to capture cohomological obstructions and δ_{k/Q,S}(A) to bound the number of relations. By constructing free and admissible Γ-presentations, establishing pro-𝒞‑level structures, and bounding multiplicities via H^1 and H^2, the authors prove finite, balanced-type presentations exist for G_O,∞(k)^𝒞 in many settings, and they extend non-abelian Cohen-Lenstra heuristics to these canonical quotients. They also analyze exceptional cases where root-of-unity phenomena alter the distributions, clarifying when the LWZB random-group model remains valid and when it requires corrections. Overall, the work provides a robust, cohomology-driven toolkit for understanding Galois groups with restricted ramification and their relationship to probabilistic models of class-field theoretic structures.

Abstract

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We prove a generalized version of the global Euler-Poincaré Characteristic, and define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch about the pro-$\ell$ completion of $G_S(k)$ to study the whole group $G_S(k)$. In the setting of the nonabelian Cohen-Lenstra heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown conjecture are always achievable by the random group that is constructed in the definition the probability measure in the conjecture.

Presentations of Galois groups of maximal extensions with restricted ramification

TL;DR

The paper develops a comprehensive cohomological framework to analyze presentations of G_S(k), the Galois group of the maximal extension unramified outside S, across global fields and Γ-extensions. It generalizes the Euler-Poincaré characteristic to include Γ-actions and arbitrary finite simples A, introducing B_S(k,A) to capture cohomological obstructions and δ_{k/Q,S}(A) to bound the number of relations. By constructing free and admissible Γ-presentations, establishing pro-𝒞‑level structures, and bounding multiplicities via H^1 and H^2, the authors prove finite, balanced-type presentations exist for G_O,∞(k)^𝒞 in many settings, and they extend non-abelian Cohen-Lenstra heuristics to these canonical quotients. They also analyze exceptional cases where root-of-unity phenomena alter the distributions, clarifying when the LWZB random-group model remains valid and when it requires corrections. Overall, the work provides a robust, cohomology-driven toolkit for understanding Galois groups with restricted ramification and their relationship to probabilistic models of class-field theoretic structures.

Abstract

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group of the maximal extension of a global field that is unramified outside a finite set of places, as varies among a certain family of extensions of a fixed global field . We prove a generalized version of the global Euler-Poincaré Characteristic, and define a group , for each finite simple -module , to generalize the work of Koch about the pro- completion of to study the whole group . In the setting of the nonabelian Cohen-Lenstra heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown conjecture are always achievable by the random group that is constructed in the definition the probability measure in the conjecture.

Paper Structure

This paper contains 25 sections, 40 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Method of the proof
  3. Previous works
  4. Other applications and further questions
  5. Notation and Preliminary
  6. Profinite groups and modules
  7. Galois groups and Galois cohomology
  8. Presentations of finitely generated profinite $\Gamma$-groups
  9. Presentations of finitely generated profinite admissible $\Gamma$-groups
  10. Presentations of finitely generated profinite $\Gamma$-groups of level ${\mathcal{C}}$
  11. The heights of pro-${\mathcal{C}}$ groups
  12. A Generalized Version of Global Euler-Poincaré Characteristic Formula
  13. Preparation for the proof
  14. Proof of Theorem \ref{['thm:EPChar']}
  15. Definition and properties of $\mathop{\mathrm{\hbox{\fontencoding{OT2} B}}}\nolimits_S(k, A)$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a nontrivial finite group and $Q$ be either ${\mathbb Q}$ or ${\mathbb F}_q(t)$ with $q$ relatively prime to $|\Gamma|$. Let ${\mathcal{C}}$ be a finite set of isomorphism classes of finite $\Gamma$-groups all of whose orders are prime to $|\mu(Q)||\Gamma|$ and $\mathop{\mathrm{char} for some positive integer $n$ and some set $X$ consisting of $n+1$ elements of ${\mathcal{F}}_n(\Ga

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Proposition 3.4
  • Remark 3.5
  • proof
  • Definition 3.6
  • ...and 82 more