Bounded generation and commutator width of Chevalley groups: function case

Boris Kunyavskiĭ; Eugene Plotkin; Nikolai Vavilov

doi:10.1007/s40879-023-00627-y

Bounded generation and commutator width of Chevalley groups: function case

Boris Kunyavskiĭ, Eugene Plotkin, Nikolai Vavilov

Department of Mathematics

St. Petersburg State University

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We prove that Chevalley groups over polynomial rings F_q[t] and over Laurent polynomial F_q[t, t^{- 1}] rings, where 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.

Original language	English
Article number	53
Journal	European Journal of Mathematics
Volume	9
Issue number	3
DOIs	https://doi.org/10.1007/s40879-023-00627-y
State	Published - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Funding

Research of Boris Kunyavskiĭ and Eugene Plotkin was supported by the ISF Grants 1623/16 and 1994/20. Nikolai Vavilov thanks the “Basis” Foundation Grant N. 20-7-1-27-1 “Higher Symbols in Algebraic K-Theory”

Funders	Funder number
Israel Science Foundation	1623/16, 1994/20

Keywords

Bounded generation
Chevalley groups
First order rigidity
Kac–Moody groups
Polynomial rings

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10.1007/s40879-023-00627-y

Cite this

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AB - We prove that Chevalley groups over polynomial rings Fq[t] and over Laurent polynomial Fq[t, t- 1] rings, where Fq 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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Bounded generation and commutator width of Chevalley groups: function case

Abstract

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Keywords

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