{
  "abstract": "We prove that if a group $G$ contains a Baumslag-Solitar group $\\mathrm{BS}(m,n)$, with $mn\\neq0$, then the theory of fields with a $G$-action has no model companion. In particular, every group containing $\\mathbb{Z}^2\\cong\\mathrm{BS}(1,1)$ has no model companion for its field actions, resolving a conjecture of Beyarslan and Kowalski. Consequently, there are no model companions for field actions of $\\mathbb{Q}^2$, Thompson's groups $F,T,V$, or Higman's group. Our argument adapts Hrushovski's method for two commuting automorphisms. The key modification is the cyclotomic condition $θ_q$, which removes the need to prescribe a non-trivial action on a root of unity and thereby makes Hrushovski's method applicable after passing from a subgroup to an ambient group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01996",
  "author_total": 1,
  "authors": [
    {
      "name": "Makoto Yanagawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xga07zn2e0rz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01996",
  "fulltext_url": null,
  "id": "5xga07zn2e0rz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01996",
  "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.GR"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7207743499",
  "pdf_url": "https://arxiv.org/pdf/2609.01996",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xga07zn2e0rz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01996",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01996"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions",
  "title_text": "Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/5xga07zn2e0rz",
  "venue": "arXiv",
  "versions": []
}