Bounded generation and commutator width of Chevalley groups: function case

2022/04/22 by Boris Kunyavskiı̆, Eugene Plotkin, Kunyavskii, Boris +3
Computer Science · Mathematics · #20G07 Structure theory for linear algebraic groups #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2204.10951

openalex publication_date 2022/04/22 · openalex created_date 2022/04/28 · openalex updated_date 2026/07/28

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.

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