Uniform bounded elementary generation of Chevalley groups

TL;DR

This paper proves that Chevalley groups of rank at least 2 over Dedekind rings of arithmetic type have a uniform bound on their elementary generation width, nearly resolving a longstanding problem in algebraic group theory.

Contribution

It establishes a universal bound on elementary generation for Chevalley groups over Dedekind rings, extending previous results to a broader class of rings and root systems.

Findings

Existence of a universal bound L for elementary width

Applicable to all Dedekind rings of arithmetic type

Covers all reduced irreducible root systems of rank ≥ 2

Abstract

In this paper we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings $R$ of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system $\Phi$ of rank $\ge 2$ there exists a universal bound $L=L(\Phi)$ such that the simply connected Chevalley groups $G(\Phi,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type $R$.