Commutator width in Chevalley groups

TL;DR

This paper investigates the commutator width in Chevalley groups, revealing that they rarely have finite commutator width due to the scarcity of commutators, and discusses related background, methods, and potential generalizations.

Contribution

It establishes that elementary Chevalley groups almost never have finite commutator width, providing new insights into their algebraic structure and generation properties.

Findings

Chevalley groups rarely have finite commutator width

Commutators have finite width in elementary generators

Discussion of background, methods, and generalizations

Abstract

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.