{
  "id": "1911.04296",
  "version": "v1",
  "published": "2019-11-11T14:17:01.000Z",
  "updated": "2019-11-11T14:17:01.000Z",
  "title": "Cardinal invariants and convergence properties of locally minimal groups",
  "authors": [
    "Dikran Dikranjan",
    "Dmitri Shakhmatov"
  ],
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that |S|=w(G)=w(K). Items (i)--(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we construct a Frechet-Urysohn minimal group G of character kappa such that the connected component of G is an open normal omega-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-11T14:17:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22A05",
      "22C05",
      "22D05",
      "54H11"
    ],
    "keywords": [
      "locally minimal groups",
      "convergence properties",
      "cardinal invariants",
      "abelian group",
      "open normal countably compact subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}