{
  "abstract": "For $n\\geq 3$, let \\[ Γ(Q)=\\langle E_{ij}(q_{ij}):i\\neq j\\rangle \\] be the subgroup of $\\operatorname{SL}(n,\\mathbb R)$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $Γ(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective. We also prove strong small-generation results for finite-index subgroups of $S$-arithmetic groups. In particular, $\\operatorname{SL}\\left(n,\\mathbb Z\\left[\\frac1N\\right]\\right)$, $n\\geq3$, and a broad class of groups $\\operatorname{SL}(2,\\mathcal O_{F,S})$ have arbitrarily small two-generated finite-index subgroups. We then study the rank-one family \\[ Γ_q=\\left\\langle\\begin{pmatrix}1&1\\\\0&1 \\end{pmatrix}, \\begin{pmatrix} 1&0\\\\q&1 \\end{pmatrix}\\right\\rangle,\\qquad q=\\tfrac{s}{t}\\in\\mathbb Q. \\] For $q\\neq0,\\pm3$, we prove that \\[ Γ_q=Γ_1^{(t)}(s) \\] if and only if its upper-triangular subgroup strictly contains \\[\\left\\langle\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\right\\rangle.\\] We then apply the rank-one criterion to constructions arising from non-freeness, verifying it for $q=\\tfrac{s}{t}\\in(-4,4)$ with $1\\leq |s|\\leq21$, and obtaining infinite families from indefinite binary quadratic forms and Pell-type equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04580",
  "author_total": 1,
  "authors": [
    {
      "name": "Yanlong Hao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0qa7p5bcra12a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04580",
  "fulltext_url": null,
  "id": "0qa7p5bcra12a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04580",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7167582954",
  "pdf_url": "https://arxiv.org/pdf/2607.04580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0qa7p5bcra12a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.04580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Groups Generated by Root Unipotents: Higher-rank and rank-one",
  "title_text": "Groups Generated by Root Unipotents: Higher-rank and rank-one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0qa7p5bcra12a",
  "venue": "arXiv",
  "versions": []
}