{
  "abstract": "We construct an explicit infinite virtually free group $G=C_4\\ast_{C_2}D_3$ with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of $G$ has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a $k$-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that $G$ is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic $k$-geodetic group is geodetic cannot both hold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04491",
  "author_total": 1,
  "authors": [
    {
      "name": "André Carvalho"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5x2jnjbrkczz7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04491",
  "fulltext_url": null,
  "id": "5x2jnjbrkczz7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04491",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220666964",
  "pdf_url": "https://arxiv.org/pdf/2610.04491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5x2jnjbrkczz7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A non-plain virtually free group with a strictly 2-geodetic Cayley graph",
  "title_text": "A non-plain virtually free group with a strictly 2-geodetic Cayley graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5x2jnjbrkczz7",
  "venue": "arXiv",
  "versions": []
}