{
  "abstract": "Let $F$ be the free group of rank two and, for a prime $p$, let $\\mathbf H_p=\\mathbf G_p*\\mathbf{Ab}_{p-1}$ be the pseudovariety of finite groups having a normal $p$-subgroup with abelian quotient of exponent dividing $p-1$. For $H$ of rank two with $\\ab_F(H)=F^{\\ab}$ we determine the set $\\mathfrak D(H)$ of primes $p$ where $H$ is $\\mathbf H_p$-dense in $F$, and we show that this prime set is Frobenian in the sense of Serre: it is governed by a single Laurent polynomial $g$ attached to $H$. We prove that $p\\in\\mathfrak D(H)$ if and only if $g$ has no zero on the torus $(\\F_p^\\times)^2$, and hence that $\\mathfrak D(H)$ possesses a computable natural density $d(H)$. Exactly one of three cases holds: $\\mathfrak D(H)$ is the set of all primes, it is finite, or it is neither and $d(H)$ satisfies $\\frac1{|G|}\\le d(H)\\le1-\\frac1{|G|}$, where $G$ is the finite Galois group attached to $g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28941",
  "author_total": 1,
  "authors": [
    {
      "name": "Jianchun Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm1sgr8y5bppx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28941",
  "fulltext_url": null,
  "id": "bm1sgr8y5bppx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28941",
  "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.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214344114",
  "pdf_url": "https://arxiv.org/pdf/2609.28941",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bm1sgr8y5bppx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28941",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28941"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pro-$\\mathbf H_p$ density in the free group of rank two: a Frobenian criterion on the mod-$p$ torus",
  "title_text": "Pro-H_p density in the free group of rank two: a Frobenian criterion on the mod-p torus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/bm1sgr8y5bppx",
  "venue": "arXiv",
  "versions": []
}