{
  "abstract": "For any infinite one-relator group $Γ=\\langle S \\mid w^m\\rangle$, we prove that $\\mathrm{cost}(Γ)=|S|-\\frac{1}{m}$. For such groups, this gives $β^{(2)}_1(Γ)=\\mathrm{cost}(Γ) - 1$, answering a special case of Gaboriau's question on the relationship between cost and first $\\ell^2$-Betti number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.23506",
  "author_total": 2,
  "authors": [
    {
      "name": "Antoine Poulin"
    },
    {
      "name": "Konrad Wróbel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dsd07txgb61c4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.23506",
  "fulltext_url": null,
  "id": "dsd07txgb61c4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.23506",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
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      "openalex",
      "2608|Geometric and Algebraic Topology"
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  "openalex": "W7165619031",
  "pdf_url": "https://arxiv.org/pdf/2606.23506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dsd07txgb61c4.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:13Z",
      "native_id": "2606.23506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.23506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Cost of one-relator groups",
  "title_text": "Cost of one-relator groups",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:13Z",
  "url": "https://research.draconitic.com/paper/dsd07txgb61c4",
  "venue": "arXiv",
  "versions": []
}