Year: 2025
Type: article
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.
Source: Algebra & Number Theory
Author: Yuan Liu
Institution: University of Illinois Urbana-Champaign
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: National Science Foundation
OpenAlex ID: W4287778587