{
  "id": "1908.03816",
  "version": "v1",
  "published": "2019-08-10T21:55:37.000Z",
  "updated": "2019-08-10T21:55:37.000Z",
  "title": "Automorphisms of the generalised Thompson's group $T_{n,r}$",
  "authors": [
    "Feyishayo Olukoya"
  ],
  "comment": "43 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The recent paper \"The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups $G_{n,r}$\" of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as the specific subgroup of the rational group $\\mathcal{R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanski{\\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of BCMNO and characterise the automorphism group of $T_{n,r}$ as a subgroup of $\\mathrm{Aut}{G_{n,r}}$. We then show that the groups $\\mathrm{Out}{T_{n,r}}$ can be identified with subgroups of the group $\\mathrm{Out}{T_{n,n-1}}$. Extending results of Brin and Guzman, we show that the groups $\\mathrm{Out}{T_{n,r}}$, for $n>2$, are all infinite and contain an isomorphic copy of Thompson's group $F$. For $X \\in \\{T,G\\}$, we study the groups $\\mathrm{Out}{X_{n,r}}$ and show that these fit in a lattice structure where $\\mathrm{Out}{X_{n,1}} \\unlhd \\mathrm{Out}{X_{n,r}}$ for all $1 \\le r \\le n-1$ and $\\mathrm{Out}{X_{n,r}} \\unlhd \\mathrm{Out}{X_{n,n-1}}$. This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of $\\mathrm{Out}{G_{n,n-1}}$. Furthermore, we deduce that for $1\\le j,d \\le n-1$ such that $d = \\gcd(j, n-1)$, $\\mathrm{Out}{X_{n,j}} = \\mathrm{Out}{X_{n,d}}$ extending a result of BCMNO for the groups $G_{n,r}$ to the groups $T_{n,r}$. We give a negative answer to the question in BCMNO which asks whether or not $\\mathrm{Out}{G_{n,r}} \\cong \\mathrm{Out}{G_{n,s}}$ if and only if $\\gcd(n-1,r) = \\gcd(n-1,s)$. We conclude by showing that the groups $T_{n,r}$ have the $R_{\\infty}$ property extending the result of Burillo, Matucci and Ventura and, independently, Gon{\\c c}alves and Sankaran, for Thompson's group $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-10T21:55:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E36",
      "20E45",
      "20E07"
    ],
    "keywords": [
      "generalised thompsons group",
      "automorphism",
      "normal subgroup structure",
      "higman groups",
      "higman-thompson groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}