{
  "id": "2309.06785",
  "version": "v1",
  "published": "2023-09-13T08:18:53.000Z",
  "updated": "2023-09-13T08:18:53.000Z",
  "title": "Key subgroups and co-minimality in topological groups",
  "authors": [
    "Michael Megrelishvili",
    "Menachem Shlossberg"
  ],
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "We introduce a minimality property for subgroups of topological groups. A subgroup $H$ is a key subgroup of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ the original topology. In fact, this concept appears implicitly in some earlier publications [9,3,11]. Every co-minimal subgroup is a key subgroup while the converse is not true even for discrete groups. In this paper, we continue the study of minimality in topological matrix groups initiated in [11]. Extending some results from [9,3] concerning the generalized Heisenberg group, we prove that the center of the upper unitriangular group $\\operatorname{UT}(n,K)$, defined over a commutative topological unital ring $K$, is a key subgroup. This center is even co-minimal in $\\operatorname{UT}(n,K)$ assuming that the multiplication map $m \\colon K\\times K\\to K$ is strongly minimal. We show that the latter holds when $K$ is an archimedean absolute valued field or a local field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-13T08:18:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological group",
      "strictly coarser hausdorff group topology",
      "co-minimality",
      "archimedean absolute valued field",
      "upper unitriangular group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}