Bounded generation and commutator width of Chevalley groups: function case

TL;DR

This paper establishes bounded elementary generation and explicit bounds on commutator width for Chevalley groups over polynomial and Laurent polynomial rings over finite fields, with implications for affine Kac--Moody groups.

Contribution

It provides the first explicit bounds on commutator width for these Chevalley groups over polynomial rings and extends results to affine Kac--Moody groups over finite fields.

Findings

Chevalley groups over polynomial rings are boundedly elementarily generated.

Explicit bounds for the commutator width of these groups are derived.

Results have applications to affine Kac--Moody groups over finite fields.

Abstract

We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit bounds of the commutator width of these groups. Under some additional assumptions, we prove similar results for other classes of Chevalley groups over Dedekind rings of arithmetic rings in positive characteristic. As a corollary, we produce explicit estimates for the commutator width of affine Kac--Moody groups defined over finite fields. The paper contains also a broader discussion of the bounded generation problem for groups of Lie type, some applications and a list of unsolved problems in the field.