{
  "abstract": "We study the elementary theories and algorithmic properties of one-relator groups. We give a first-order sentence characterizing Baumslag-Solitar-freeness in this class. For fixed $|m|>1$, we regularly interpret $\\BS(1,m)$ as a definable subgroup in every one-relator group containing it. We uniformly, regularly and injectively interpret integer arithmetic in countable groups of cohomological dimension at most two. We obtain one uniform interpretation for all one-relator groups containing a non-unimodular Baumslag--Solitar subgroup. Every fixed finitely presented group is regularly interpretable in these groups, and their elementary theories are undecidable. We also describe the GBS overgroups of a non-unimodular Baumslag--Solitar subgroup. For the unimodular groups we construct finite-index direct products of free groups and $\\Z$, and reduce the elementary theory of $\\BS(k,k)$ to that of a free group with a named periodic automorphism. Using the recent Open AI result that BS-free one-relator groups are hyperbolic we obtain a terminating algorithm recognizing this class and obtain consequences. From the virtual compact specialness result of another Open AI paper, we obtain an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel may have infinite rank. For classes of torsion-free one-relator groups in which CSA implies toral relative hyperbolicity, we give a uniform algorithm deciding CSA and, in the affirmative case, the existence of a subgroup isomorphic to $\\Z^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25305",
  "author_total": 2,
  "authors": [
    {
      "name": "Olga Kharlampovich"
    },
    {
      "name": "Alexei Miasnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40z1nss0yc77c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25305",
  "fulltext_url": null,
  "id": "40z1nss0yc77c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25305",
  "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": "W7223023522",
  "pdf_url": "https://arxiv.org/pdf/2610.25305",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40z1nss0yc77c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25305",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25305"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Model Theory and Algorithms for One-Relator Groups",
  "title_text": "Model Theory and Algorithms for One-Relator Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/40z1nss0yc77c",
  "venue": "arXiv",
  "versions": []
}