{
  "abstract": "Let $T=\\langle f,g\\mid f^{l}=g^{m}=W^{n}=1\\rangle$ be a generalised triangle group in which $W$ is the Farey word of a rational slope $q=r/s$, and suppose that $T$ is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let $G=\\langle f,g\\mid f^{l}=g^{m}=W^{n}\\rangle$ be the associated central extension. We compute an explicit free resolution of the trivial module over $\\mathbb{Z}G$ of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over $\\mathbb{Z}T$ which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family $q=0/1$ in 1983. Since then only the family $q=1/2$ has been treated, by Ellis and Williams, and for that family our resolution over $\\mathbb{Z}T$ has strictly smaller rank in every degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20932",
  "author_total": 3,
  "authors": [
    {
      "name": "Sean P. Carroll"
    },
    {
      "name": "Thomas Csizmadia"
    },
    {
      "name": "Jeroen Schillewaert"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/87r5v243dj1cr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20932",
  "fulltext_url": null,
  "id": "87r5v243dj1cr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20932",
  "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.AT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7222761768",
  "pdf_url": "https://arxiv.org/pdf/2610.20932",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/87r5v243dj1cr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20932",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20932"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Resolutions of orbifold generalised triangle groups",
  "title_text": "Resolutions of orbifold generalised triangle groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/87r5v243dj1cr",
  "venue": "arXiv",
  "versions": []
}