{
  "abstract": "We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, $\\#_7(S^2\\times S^2)$ is topologically unrecognizable and $\\#_9(S^2\\times S^2)$ is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10461",
  "author_total": 3,
  "authors": [
    {
      "name": "Marc Kegel"
    },
    {
      "name": "Shana Yunsheng Li"
    },
    {
      "name": "Qiuyu Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8wvt77bmjck09.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10461",
  "fulltext_url": null,
  "id": "8wvt77bmjck09",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10461",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212124734",
  "pdf_url": "https://arxiv.org/pdf/2609.10461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8wvt77bmjck09.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:16Z",
      "native_id": "2609.10461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10461"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Small undecidable groups and unrecognizable 4-manifolds",
  "title_text": "Small undecidable groups and unrecognizable 4-manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/8wvt77bmjck09",
  "venue": "arXiv",
  "versions": []
}