{
  "abstract": "A tubular group has a graph of groups decomposition with $\\mathbb{Z}^2$ vertex groups and $\\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\\ell),(m,n)} = \\langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\\ell \\rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number\" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05705",
  "author_total": 1,
  "authors": [
    {
      "name": "Amy Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51x0yx50rzct9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05705",
  "fulltext_url": null,
  "id": "51x0yx50rzct9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05705",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212058993",
  "pdf_url": "https://arxiv.org/pdf/2609.05705",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51x0yx50rzct9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05705",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05705"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Commensurability and quasi-isometry classification for one vertex one loop tubular groups",
  "title_text": "Commensurability and quasi-isometry classification for one vertex one loop tubular groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/51x0yx50rzct9",
  "venue": "arXiv",
  "versions": []
}