{
  "abstract": "We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an \\(F(S)\\)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator \\(w^m\\), cyclic translates of the repaired set amplify a zero-defect set of measure close to \\(1/m\\) to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02213",
  "author_total": 1,
  "authors": [
    {
      "name": "Jinhyuk Son"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b26qv25kf2m3n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02213",
  "fulltext_url": null,
  "id": "b26qv25kf2m3n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02213",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220379198",
  "pdf_url": "https://arxiv.org/pdf/2610.02213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b26qv25kf2m3n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.02213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Soficity of One-Relator Groups and Reducible Presentations",
  "title_text": "Soficity of One-Relator Groups and Reducible Presentations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/b26qv25kf2m3n",
  "venue": "arXiv",
  "versions": []
}