{
  "id": "2304.02504",
  "version": "v1",
  "published": "2023-04-05T15:24:18.000Z",
  "updated": "2023-04-05T15:24:18.000Z",
  "title": "Finite axiomatizability of the rank and the dimension of a pro-$π$ group",
  "authors": [
    "Martina Conte",
    "Benjamin Klopsch"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "The Pr\\\"ufer rank $\\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits the structure of a $p$-adic analytic group if and only if $G$ is virtually a pro-$p$ group of finite rank. The dimension $\\dim G$ of a $p$-adic analytic profinite group $G$ is the analytic dimension of $G$ as a $p$-adic manifold; it is known that $\\dim G$ coincides with the rank $\\mathrm{rk}(U)$ of any uniformly powerful open pro-$p$ subgroup $U$ of $G$. Let $\\pi$ be a finite set of primes, let $r \\in \\mathbb{N}$ and let $\\mathbf{r} = (r_p)_{p \\in \\pi}, \\mathbf{d} = (d_p)_{p \\in \\pi}$ be tuples in $\\{0, 1, \\ldots,r\\}$. We show that there is a single sentence $\\sigma_{\\pi,r,\\mathbf{r},\\mathbf{d}}$ in the first-order language of groups such that for every pro-$\\pi$ group $G$ the following are equivalent: (i) $\\sigma_{\\pi,r,\\mathbf{r},\\mathbf{d}}$ holds true in the group $G$, that is, $G \\models \\sigma_{\\pi,r,\\mathbf{r},\\mathbf{d}}$; (ii) $G$ has rank $r$ and, for each $p \\in \\pi$, the Sylow pro-$p$ subgroups of $G$ have rank $r_p$ and dimension $d_p$. Loosely speaking, this shows that, for a pro-$\\pi$ group $G$ of bounded rank, the precise rank of $G$ as well as the ranks and dimensions of the Sylow subgroups of $G$ can be recognized by a single sentence in the first-order language of groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-05T15:24:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "03C98",
      "20A15",
      "20D15",
      "20D20",
      "22E20"
    ],
    "keywords": [
      "finite axiomatizability",
      "first-order language",
      "single sentence",
      "adic analytic profinite group",
      "adic analytic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}