{
  "id": "1910.01081",
  "version": "v1",
  "published": "2019-10-02T16:56:04.000Z",
  "updated": "2019-10-02T16:56:04.000Z",
  "title": "On the minimal number of generators of endomorphism monoids of full shifts",
  "authors": [
    "Alonso Castillo-Ramirez"
  ],
  "comment": "Extended version of arXiv:1901.02808",
  "categories": [
    "math.GR",
    "math.DS"
  ],
  "abstract": "For a group $G$ and a finite set $A$, denote by $\\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the \\emph{rank}, of $\\text{End}(A^G)$ and $\\text{Aut}(A^G)$. In the first part, when $G$ is a finite group, we give upper and lower bounds for the rank of $\\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of $G$. In the second part, we apply our bounds to show that if $G$ has an infinite descending chain of normal subgroups of finite index, then $\\text{End}(A^G)$ is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-02T16:56:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "68Q80",
      "05E18",
      "20M20"
    ],
    "keywords": [
      "minimal number",
      "endomorphism monoids",
      "full shifts",
      "generators",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}