{
  "abstract": "Let $\\mathcal{O}_S$ 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 $|S| \\ge 2$, then every unimodular $(2 \\times 2)$-matrix over $\\mathcal{O}_S$ is a product of $\\leq 7$ elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of $SL_2(\\mathcal{O}_S)$ started over 50 years ago.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.21817",
  "author_total": 3,
  "authors": [
    {
      "name": "B. E. Kunyavskii"
    },
    {
      "name": "D. W. Morris"
    },
    {
      "name": "A. S. Rapinchuk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aptd4krypp9az.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.21817",
  "fulltext_url": null,
  "id": "aptd4krypp9az",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.21817",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7165628711",
  "pdf_url": "https://arxiv.org/pdf/2606.21817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aptd4krypp9az.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2606.21817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.21817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Bounded elementary generation of SL_2: nearly the end",
  "title_text": "Bounded elementary generation of SL_2: nearly the end",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/aptd4krypp9az",
  "venue": "arXiv",
  "versions": []
}