{
  "abstract": "We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group $G$ of cardinality $κ$, we construct $2^κ$ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and $κ$ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01049",
  "author_total": 1,
  "authors": [
    {
      "name": "Alex J. Sutherland"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqrnacch2mj9g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01049",
  "fulltext_url": null,
  "id": "eqrnacch2mj9g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01049",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7216578727",
  "pdf_url": "https://arxiv.org/pdf/2610.01049",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqrnacch2mj9g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.01049",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01049"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Watkins's conjecture holds for all infinite groups",
  "title_text": "Watkins's conjecture holds for all infinite groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/eqrnacch2mj9g",
  "venue": "arXiv",
  "versions": []
}