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Uniform bounded elementary generation of Chevalley groups

Boris Kunyavskii, Eugene Plotkin, Nikolai Vavilov

Abstract

In this paper we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings $R$ of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system $Φ$ of rank $\ge 2$ there exists a universal bound $L=L(Φ)$ such that the simply connected Chevalley groups $G(Φ,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type $R$.

Uniform bounded elementary generation of Chevalley groups

Abstract

In this paper we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank over arbitrary Dedekind rings of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system of rank there exists a universal bound such that the simply connected Chevalley groups have elementary width for all Dedekind rings of arithmetic type .
Paper Structure (23 sections, 29 theorems, 69 equations)

This paper contains 23 sections, 29 theorems, 69 equations.

Table of Contents

  1. Notation and preliminaries
  2. Chevalley groups
  3. Relative groups
  4. Arithmetic case
  5. Supporting statements
  6. Reduction to the ring of integers
  7. Arithmetic lemmas
  8. Tavgen rank reduction theorem and applications
  9. Tavgen rank reduction theorem
  10. The case when $R^*$ is infinite
  11. The simply laced case and $\Phi=\mathsf F_4$
  12. Stability of K$_1$-functor and flipping long and short roots
  13. The case of $\mathrm{Sp}(2l,R)$, $l\ge 3$
  14. Swindling lemma
  15. Swindling lemma for $\mathrm{SL}(3,R)$
  16. ...and 8 more sections

Key Result

Theorem A

Let $\Phi$ be a reduced irreducible root system of rank $l\ge 2$. Then there exists a constant $L=L(\Phi)$, depending on $\Phi$ alone, such that for any Dedekind ring of arithmetic type $R$, any element in $G_{\mathrm{sc}}(\Phi,R)$ is a product of at most $L$ elementary root unipotents,

Theorems & Definitions (51)

  • Theorem A
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem B
  • Lemma 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 41 more