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On the length of commutators in Chevalley groups

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Abstract

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A l , with a better bound, was earlier obtained by Sivatski and Stepanov.

The width of the linear elementary group E n (R) with repect to elementary generators or the set of all commutators was studied by Carter, Keller, Dennis, Vaserstein, van der Kallen and others. It is related to the computation of the Kazhdan constant. For example, the width of (E n (R) is finite if R is semilocal (by Gauus decomposition) of R = ℤ (Carter, Keller), but is infinite for R = ℂ[x] (van der Kallen). Already for R = F[x] over a finite field F this question is open.

Van der Kallen noticed that the group E(Φ, R)∞/E(Φ,R ∞) is an obstruction for the finitness of width of E(Φ, R), were infinte power means the direct product of countably many copies of a ring or a group. The main resutl of the article is equivalent to the fact that this group is central in K 1(Φ,R ∞).

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Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences at GCU, Lahore, Pakistan

    Alexei Stepanov

  2. Sankt-Petersburg Electrotechnical University, 197342, Sankt-Petersburg, Russia

    Alexei Stepanov

  3. Department of Mathematics and Mechanics, Sankt-Petersburg State University, 198904, Petrodvorets, Russia

    Nikolai Vavilov

Authors
  1. Alexei Stepanov
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  2. Nikolai Vavilov
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Correspondence to Alexei Stepanov.

Additional information

At various stages the work on this article was supported by INTAS 03-51-3251, RFFI 03-01-00349, NSh-8464.2006.1 (Russian Ministry of Education), and RFFI research projects 08-01-00756 (RGPU), 09-01-00762 (Siberian Federal University), 09-01-00784 (POMI), 09-01-00878 (SPbGU), RFFI-DFG cooperation project 09-01-91333 (POMI-LMU) and RFFI-VAN cooperation project 09-01-90304 (SPbGU-HCMU).

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Stepanov, A., Vavilov, N. On the length of commutators in Chevalley groups. Isr. J. Math. 185, 253–276 (2011). https://doi.org/10.1007/s11856-011-0109-2

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