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Bounded generation and commutator width of Chevalley groups: function case

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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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Notes

  1. The difference between the number and function cases is subtle enough and may be overlooked when approaching from outside. We quote from page 2 of the memoir [28]: ‘...G is known to be boundedly generated by X only in a few cases, namely, when R is a finite extension of \({\mathbb {Z}}\) or F[t], with F a finite field.’ In a sense, the present paper, along with [51], can be viewed as a first step along the long and painful road to justification of this brave claim.

  2. After the preliminary version of the present paper has been finished, there appeared a preprint of Alexander Trost [81] where the statement of our Theorem A was established for the ring of integers R of an arbitrary global function field K, with a bound of the form , where the factor L depends on q and of the degree d of K. His method is similar to Morris’ approach in [49]. The subsequent preprint [82] contains further improvements for the group , \(n\geqslant 3\), over an arbitrary global function ring R.

  3. On the other hand, since \(B\cap U^-=e\), it is never a product of three such unipotents, unless \(\varepsilon =1\).

  4. In the literature, expressions bounded generation and finite width are used in several related but significantly different contexts. Oftentimes one calls a group G boundedly generated if it has bounded generation with respect to the powers of some finite generating set. This amounts to the group being a finite product of several cyclic subgroups. In many situations it is equally meaningful to consider groups which are finite products of abelian subgroups. Finally, one calls families \(G_i\) of finitely presented groups boundedly generated if they can be presented in such a way that the sums of the number of generators and relations of \(G_i\) are uniformly bounded.

  5. In the literature, three of four completely different commodities are merchandised under the common name of Gauß decomposition: (1) the big cell decomposition LU, as in the affine group schemes textbooks, (2) the Birkhoff LPU-decomposition as in Ellers–Gordeev [25], (3) the LUP-decomposition, as in the computational linear algebra textbooks. Here we speak of (4) the DULU-decomposition, consult [68] for the historical background.

  6. Or, actually, the number of its prime divisors. Later, Loukanidis and Murty [43, 50] obtained bounds that depended on n and the degree \(|K\,{:}\,\mathbb {Q}|\) of K, not the discriminant.

  7. Well, explicit use of infinitesimals does make life easier, sometimes. Say, in R itself there are no non-obvious ideals such that \(I^2=I\), whereas in \({}^*R\) there is such an ideal consisting of all infinitesimal elements, which can be very handy. But these simplifications are mostly relevant in the [difficult] case \(n=2\), see the next subsection.

  8. The proof from [10] with several successful deteriorations, without reference to [10], and with a worse bound 73 was subsequently reproduced in [2].

  9. See [89], where it is [essentially] proven for \(R=\mathbb {Z}[1/p]\), again modulo GRH.

  10. Recall that our standing assumption \(|S|\geqslant 2\) excludes the problematic case \(R=\mathbb {Z}\).

  11. As we know from Sect. 4, this does not influence boundedness or lack thereof, but may affect the actual bounds.

  12. There is an exception: in [9] the term ‘local units’ is used for calling nonzero elements of the local field \(K_v\), where v is a place of K. We avoid using such a terminology because ‘local unit’ commonly refers to an invertible element of the valuation ring \(O_v\).

  13. The relevant facts in [14] are formulated for Kac–Moody groups over \({\mathbb {C}}\). However, the construction remains valid for an appropriate \({\mathbb {Z}}\)-model [30] and hence the needed results from [14] can be extended to groups over \({\mathbb {F}}_q\).

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Acknowledgements

Very special thanks go to Inna Capdeboscq. This paper started jointly with her as a discussion of the bounded commutator width of various classes of Kac–Moody groups over finite fields, and for a long time it was supposed to be a work of four authors. Our sincere thanks go to Nikolai Gordeev, Olga Kharlampovich, Jun Morita, Alexei Myasnikov, Denis Osipov, Alexei Stepanov, and Igor Zhukov for useful discussions regarding various aspects of this work. We thank the referees for careful reading and meticulous remarks.

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  1. Department of Mathematics, Bar-Ilan University, Ramat Gan, 5290002, Israel

    Boris Kunyavskiĭ & Eugene Plotkin

  2. Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia

    Nikolai Vavilov

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Kunyavskiĭ, B., Plotkin, E. & Vavilov, N. Bounded generation and commutator width of Chevalley groups: function case. European Journal of Mathematics 9, 53 (2023). https://doi.org/10.1007/s40879-023-00627-y

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