{
  "id": "2309.17065",
  "version": "v1",
  "published": "2023-09-29T08:55:09.000Z",
  "updated": "2023-09-29T08:55:09.000Z",
  "title": "More on $z$-minimality of topological groups",
  "authors": [
    "Dekui Peng",
    "Menachem Shlossberg"
  ],
  "categories": [
    "math.GN",
    "math.GR",
    "math.NT"
  ],
  "abstract": "The authors of [4] introduced the notion of $z$-minimality. A minimal group $G$ is $z$-minimal if $G/Z(G)$ is minimal. Answering a question of G. Luk\\'acs, we present, in this paper, the $z$-Minimality Criterion for dense subgroups. This criterion implies that if $\\Bbb F$ is a subfield of a local field of characteristic distinct than $2,$ then the special linear group $\\operatorname{SL}(n,\\Bbb F)$ is $z$-minimal precisely when its subgroup $\\operatorname{ST^+}(n,\\Bbb F)$, the special upper triangular group, is $z$-minimal. Some applications to Number Theory are provided. We prove that the following conditions are equivalent for a positive integer $n$: (a) $n$ is not square-free; (b) there exists a subfield $\\mathbb{F}$ of $\\mathbb{C}$ such that $\\operatorname{SL}(n, \\mathbb{F})$ is minimal but not $z$-minimal; (c) there exists a subfield $\\mathbb{F}$ of $\\mathbb{C}$ such that $\\operatorname{ST^+}(n, \\mathbb{F})$ is minimal but not $z$-minimal. We also answer [18, Question 6] in the positive, showing that both topological products $G=\\prod_{p\\in \\mathcal P}\\operatorname{SL}(p+1,(\\mathbb{Q},\\tau_p))$ and $H=\\prod_{p\\in \\mathcal P}\\operatorname{ST^+}(p+1,(\\mathbb{Q},\\tau_p))$ are minimal, where $\\mathcal P$ is the set of all primes and $(\\mathbb{Q},\\tau_p)$ is the field of rationals equipped with the $p$-adic topology. In another direction, it is proved that if $G$ is a locally compact $z$-minimal group, then the quotient group $G/Z(G)$ is topologically isomorphic to $\\operatorname{Inn}(G),$ the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-29T08:55:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological groups",
      "minimal group",
      "special upper triangular group",
      "special linear group",
      "minimality criterion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}