{
  "abstract": "Let $(G,θ)$ be a finitely generated 1-smooth oriented pro-$p$ group with torsion-free orientation, and suppose that $G$ is a Poincaré duality pro-$p$ group of dimension three. The orientation determines a canonical closed normal subgroup $K_θ(G)$ contained in the Frattini subgroup, and for a Kummerian pair the quotient $G/K_θ(G)$ is the maximal $θ$-abelian quotient. We prove that \\[K_θ(G)\\text{ finitely generated} \\quad\\Longrightarrow\\quad K_θ(G)=1.\\] Thus every non-$θ$-abelian 1-smooth $\\textrm{PD}^3$ pro-$p$ group has an infinitely generated canonical kernel, where ``infinitely generated'' means not topologically finitely generated. The proof combines the normal-subgroup theorem for pro-$p$ $\\textrm{PD}^3$ groups with the classification of $p$-adic analytic 1-smooth pro-$p$ groups. We also obtain a cohomological description of the canonical extension from the Hochschild--Serre five-term sequence and Poincaré duality. Writing $K=K_θ(G)$, $Q=G/K$, and $V=H^1(G,\\mathbb{F}_p)$, we identify $H^1(K,\\mathbb{F}_p)^Q$ with the kernel of the map $\\bigwedge^2V\\to V^*$ induced by the triple cup-product form of $G$. Consequently, whenever $K\\neq1$, its first cohomology is infinite-dimensional, whereas its $Q$-invariant subspace is nonzero and finite-dimensional. We obtain exact dimension restrictions in small generator ranks and show that a non-$θ$-abelian group with $2\\leq d(G)\\leq4$ is not Bloch--Kato. In the finitely generated-kernel case, $G$ is a three-dimensional uniform $θ$-abelian group with exterior mod-$p$ cohomology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20915",
  "author_total": 1,
  "authors": [
    {
      "name": "Marina Palaisti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3h29wbjp6wkz5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20915",
  "fulltext_url": null,
  "id": "3h29wbjp6wkz5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20915",
  "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",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7222557207",
  "pdf_url": "https://arxiv.org/pdf/2610.20915",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3h29wbjp6wkz5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20915",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20915"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Canonical Kummerian Kernels in 1-Smooth Pro-$p$ Poincaré Duality Groups of Dimension Three",
  "title_text": "Canonical Kummerian Kernels in 1-Smooth Pro-p Poincaré Duality Groups of Dimension Three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3h29wbjp6wkz5",
  "venue": "arXiv",
  "versions": []
}