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PreprintCC BY-NC-SA 4.0arXiv

Bounded elementary generation of SL_2: nearly the end

  • B. E. Kunyavskii
  • D. W. Morris
  • A. S. Rapinchuk
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Abstract

Let 𝒪S be the ring of S-integers of a global field K of any characteristic, where S is a finite set of valuations of K (and S contains all of the archimedean valuations if the characteristic is zero). We prove that if |S| ≥ 2, then every unimodular (2 × 2)-matrix over 𝒪S is a product of ≤ 7 elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of SL2(𝒪S) started over 50 years ago.

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Cite

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BibTeX
@misc{kunyavskii2026bounded,
  title = {{Bounded elementary generation of SL_2: nearly the end}},
  author = {B. E. Kunyavskii and D. W. Morris and A. S. Rapinchuk},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.21817},
  eprint = {2606.21817},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.21817},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Bounded elementary generation of SL_2: nearly the end
AU  - B. E. Kunyavskii
AU  - D. W. Morris
AU  - A. S. Rapinchuk
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2606.21817
UR  - https://arxiv.org/abs/2606.21817
AB  - Let 𝒪_S be the ring of S-integers of a global field K of any characteristic, where S is a finite set of valuations of K (and S contains all of the archimedean valuations if the characteristic is zero). We prove that if |S| ≥ 2, then every unimodular (2 × 2)-matrix over 𝒪_S is a product of ≤ 7 elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of SL_2(𝒪_S) started over 50 years ago.
ER  -