{
  "id": "2009.05450",
  "version": "v1",
  "published": "2020-09-11T13:59:11.000Z",
  "updated": "2020-09-11T13:59:11.000Z",
  "title": "A Description of Aut(dVn) and Out(dVn) Using Transducers",
  "authors": [
    "Luke Elliott"
  ],
  "comment": "17 Pages, 4 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "The groups $dV_n$ are an infinite family of groups, first introduced by C. Mart\\'inez-P\\'erez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompson groups $V_n(=1V_n)$ and the Brin-Thompson groups $nV(=nV_2)$. A description of the groups $\\operatorname{Aut}(G_{n, r})$ (including the groups $G_{n,1}=V_n$) has previously been given by C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses the transducer representations of homeomorphisms of Cantor space introduced a paper of R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M. Rubin. We generalise the transducers of the latter paper and make use of these transducers to give a description of $\\operatorname{Aut}(dV_n)$ which extends the description of $\\operatorname{Aut}(1V_n)$ given in the former paper. We make use of this description to show that $\\operatorname{Out}(dV_2) \\cong \\operatorname{Out}(V_2)\\wr S_d$, and more generally give a natural embedding of $\\operatorname{Out}(dV_n)$ into $\\operatorname{Out}(G_{n, n-1})\\wr S_d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-11T13:59:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B27",
      "20E36",
      "37B05"
    ],
    "keywords": [
      "description",
      "higman-thompson groups",
      "brin-thompson groups",
      "transducer representations",
      "cantor space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}