On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

TL;DR

This paper establishes bounds on the size of irredundant generating sets in Lie and algebraic groups, showing that larger generating sets are necessarily redundant, with implications for related conjectures.

Contribution

It provides new quantitative bounds on generating set sizes in Lie and algebraic groups and explores redundancy under Nielsen transformations, linking to existing conjectures.

Findings

Topologically generating sets larger than a polynomial bound are redundant.

Similar bounds hold for amenable Lie groups and reductive algebraic groups.

Redundancy up to Nielsen transformations relates to the Wiegold conjecture.

Abstract

We show that a topologically generating set $S$ of a connected compact Lie group $G$ of size larger than a fixed polynomial in the rank of $G$ must be redundant (i.e., some proper subset of $S$ still topologically generates $G$). Similar results are obtained for amenable Lie groups and for reductive algebraic groups with the Zariski topology. The quantitative bounds produced by our method are controlled by corresponding bounds for finite simple groups of Lie type. We also treat redundancy up to Nielsen transformations, thereby partially answering a few conjectures of Gelander. We show that these conjectures are implied by the Wiegold conjecture.