{
  "abstract": "Let \\[1\\longrightarrow N\\longrightarrow G\\longrightarrow D\\longrightarrow1\\] be exact, with $N\\neq1$ free pro-$p$ of arbitrary rank and $D$ Demuškin. Writing $W=N/Φ(N)$, we prove \\[H^3(G,\\mathbb{F}_p)^\\vee\\simeq W^D,\\qquad \\textrm{cd}_p G=3\\text{ if and only if }W^D\\neq0.\\] Thus dimension three is detected on the first Frattini layer of the free kernel. Finite-rank kernels always satisfy the criterion, so a dimension drop requires an infinite-rank kernel, while the criterion remains stable under passage to open subgroups. On the fixed-point-free locus $W^D=0$, the remaining dimension-one/two alternative is governed by $H^1(D,W^\\vee)$ together with the Hochschild--Serre transgression. Dualizing the latter produces a relation-defect class in the topological coinvariants of $W$, which admits an explicit description in terms of a lifted defining relation of the Demuškin quotient. This yields a complete criterion for the lower-dimensional branch and separates the top-degree fixed-point obstruction from the extension data controlling dimensions one and two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00021",
  "author_total": 1,
  "authors": [
    {
      "name": "Marina Palaisti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4w762k4rtxgc0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00021",
  "fulltext_url": null,
  "id": "4w762k4rtxgc0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00021",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7216328912",
  "pdf_url": "https://arxiv.org/pdf/2610.00021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4w762k4rtxgc0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.00021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-$p$ Groups",
  "title_text": "Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-p Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4w762k4rtxgc0",
  "venue": "arXiv",
  "versions": []
}