{
  "id": "2005.01685",
  "version": "v1",
  "published": "2020-05-04T17:41:24.000Z",
  "updated": "2020-05-04T17:41:24.000Z",
  "title": "Right-angled Artin pro-$p$ groups",
  "authors": [
    "Ilir Snopce",
    "Pavel Zaleskii"
  ],
  "comment": "18 pages",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime. The right-angled Artin pro-$p$ group $G_{\\Gamma}$ associated to a fnite simplicial graph $\\Gamma$ is the pro-$p$ completion of the right-angled Artin group associated to $\\Gamma$. We prove that the following assertions are equivalent: (i) no induced subgraph of $\\Gamma$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of $G_{\\Gamma}$ is itself a right-angled Artin pro-$p$ group (possibly infinitely generated); (iii) $G_{\\Gamma}$ is a Bloch-Kato pro-$p$ group; (iv) every closed subgroup of $G_{\\Gamma}$ has torsion free abelianization; (v) $G_{\\Gamma}$ occurs as the maximal pro-$p$ Galois group $G_K(p)$ of some field $K$ containing a primitive $p$th root of unity; (vi) $G_{\\Gamma}$ can be constructed from $\\mathbb{Z}_p$ by iterating two group theoretic operations, namely, direct products with $\\mathbb{Z}_p$ and free pro-$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for right-angled Artin pro-$p$ groups. Moreover, we prove that $G_{\\Gamma}$ is coherent if and only if each circuit of $\\Gamma$ of length greater than three has a chord.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-04T17:41:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "12F10",
      "20F36",
      "20E06",
      "20E08",
      "12G05"
    ],
    "keywords": [
      "fnite simplicial graph",
      "closed subgroup",
      "torsion free abelianization",
      "group theoretic operations",
      "length greater"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}