Lifting Generators in Connected Lie Groups

TL;DR

The paper addresses lifting generating sets along open epimorphisms for connected Lie groups, proving that the problem reduces to the abelianisation via $G/\overline{G'}$ and $H/f(\overline{G'})$, with a central result that perfect connected Lie groups are Gaschütz. By solving the abelian case completely and then handling semisimple, reductive, and Abels–Noskov groups, the authors extend the Gaschütz property to all connected Lie groups and provide precise bounds for the Gaschütz rank, including explicit formulas in terms of $d(G)$, $\dim(G/\overline{G'})$, and the maximal torus. They also determine the maximal size of irredundant generating sets for connected abelian Lie groups and discuss links to the Wiegold conjecture. The work offers a comprehensive framework unifying lifting problems across Lie-theoretic structures and yields concrete, transferable criteria for when lifting is possible. These results have potential applications in group presentations, subgroup growth, and automorphism-group analyses in Lie-theoretic contexts.

Abstract

Given an epimorphism between topological groups $f:G\to H$, when can a generating set of $H$ be lifted to a generating set of $G$? We show that for connected Lie groups the problem is fundamentally abelian: generators can be lifted if and only if they can be lifted in the induced map between the abelianisations (assuming the number of generators is at least the minimal number of generators of $G$). As a consequence, we deduce that connected perfect Lie groups satisfy the Gaschütz lemma: generating sets of quotients can always be lifted. If the Lie group is not perfect, this may fail. The extent to which a group fails to satisfy the Gaschütz lemma is measured by its \emph{Gaschütz rank}, which we bound for all connected Lie groups, and compute exactly in most cases. Additionally, we compute the maximal size of an irredundant generating set of connected abelian Lie groups, and discuss connections between such generation problems with the Wiegold conjecture.

Theorems & Definitions (75)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Lemma 1.8
• Theorem 1.9
• Proposition 1.11
• Definition 2.2