Presentations of Galois groups of maximal extensions with restricted ramification

Liu, Yuan

doi:10.2140/ant.2025.19.835

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Presentations of Galois groups of maximal extensions with restricted ramification

Yuan Liu

Vol. 19 (2025), No. 5, 835–881

DOI: 10.2140/ant.2025.19.835

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 $G_{S} (k)$ , the Galois group 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 define a group $B_{S} (k, A)$ , for each finite simple $G_{S} (k)$ -module $A$ , to generalize the work of Koch and Shafarevich on the pro- $ℓ$ completion of $G_{S} (k)$ . We prove that $G_{S} (k)$ always admits a balanced presentation when it is finitely generated. In the setting of the nonabelian Cohen–Lenstra heuristics, we prove that the unramified Galois groups studied by the Liu–Wood–Zureick-Brown conjecture always admit a balanced presentation in the form of the random group in the conjecture.

Keywords

presentation of Galois groups, class groups, nonabelian Cohen–Lenstra heuristics

Mathematical Subject Classification

Primary: 11R29, 11R32, 11R34

Milestones

Received: 4 February 2023

Revised: 17 December 2023

Accepted: 16 July 2024

Published: 22 April 2025

Authors

Yuan Liu
Department of Mathematics University of Illinois Urbana-Champaign Urbana, IL United States

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Vol. 19, No. 5, 2025