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Bounded Generation for $SL_n(Λ)$

Nir Avni, Chen Meiri

TL;DR

This work proves unconditional bounded generation for $SL_d(\Lambda)$ with $d\ge 3$ when $\Lambda$ is an $S$-order in a central division algebra over a number field and $SL_3(\Lambda)$ is generated by elementary matrices. The authors reduce to the case $d=3$, and develop a framework based on $S$-congruence and commgruence completions, ultrafilter ultrapowers, and discrete metaplectic extensions. By translating a potential failure of bounded generation into the existence of a nontrivial metaplectic extension and then ruling out such extensions via metaplectic kernel arguments, they force the desired bounded generation by elementary matrices. The approach blends local–global methods, Bass–Masser–Draxl–Mor-type stability results, and central extension theory to achieve a new isotropic higher-rank bounded generation result with potential implications for width questions in arithmetic groups.

Abstract

Let $Λ$ be an order in a division algebra over a number field. We prove, under some conditions, that $SL_3(Λ)$ is boundedly generated by elementary matrices.

Bounded Generation for $SL_n(Λ)$

TL;DR

This work proves unconditional bounded generation for with when is an -order in a central division algebra over a number field and is generated by elementary matrices. The authors reduce to the case , and develop a framework based on -congruence and commgruence completions, ultrafilter ultrapowers, and discrete metaplectic extensions. By translating a potential failure of bounded generation into the existence of a nontrivial metaplectic extension and then ruling out such extensions via metaplectic kernel arguments, they force the desired bounded generation by elementary matrices. The approach blends local–global methods, Bass–Masser–Draxl–Mor-type stability results, and central extension theory to achieve a new isotropic higher-rank bounded generation result with potential implications for width questions in arithmetic groups.

Abstract

Let be an order in a division algebra over a number field. We prove, under some conditions, that is boundedly generated by elementary matrices.

Paper Structure

This paper contains 13 sections, 20 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Completions
  4. The congruence completion
  5. The commgruence completion
  6. Discrete metaplectic extensions
  7. Proof of Theorem \ref{['thm:main']}
  8. Introduction
  9. Completions
  10. The congruence completion
  11. The commgruence completion
  12. Discrete metaplectic extensions
  13. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.0.1

Let $d\ge 3$, $F$ be a number field, $S$ be a finite set of places of $F$, and $D$ be an $n^2$-dimensional central division algebra over $F$. Let $O$ be the ring of $S$-integers of $F$ and let $\Lambda \subseteq D$ be an $O$-order. Assume that: Then $\mathop{\mathrm{SL}}\nolimits_d(\Lambda)$ is boundedly generated by the elementary matrices.

Theorems & Definitions (50)

  • Theorem 1.0.1
  • Remark 1.0.2
  • Remark 1.0.3
  • Definition 2.1.1
  • Lemma 2.1.2
  • proof
  • Definition 2.2.1
  • Lemma 2.2.2
  • Definition 2.2.3
  • Definition 2.2.4
  • ...and 40 more