## 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 |
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Article number | 53 |

Journal | European Journal of Mathematics |

Volume | 9 |

Issue number | 3 |

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 |
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Israel Science Foundation | 1623/16, 1994/20 |

## Keywords

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