{
  "id": "1512.00079",
  "version": "v1",
  "published": "2015-11-30T22:42:56.000Z",
  "updated": "2015-11-30T22:42:56.000Z",
  "title": "Growth rate of endomorphisms of Houghton's groups",
  "authors": [
    "Jong Bum Lee",
    "Sang Rae Lee"
  ],
  "comment": "36 pages, 4 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "A Houghton's group $\\mathcal{H}_n$ consists of translations at infinity of a $n$ rays of discrete points on the plane. In this paper we study the growth rate of endomorphisms of Houghton's groups. We show that if the kernel of an endomorphism $\\phi$ is not trivial then the growth rate $\\mathrm{GR}(\\phi)$ equals either $1$ or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism $\\phi$ of $\\mathcal{H}_n$ determines a unique natural number $\\ell$ such that $\\phi(\\mathcal{H}_n)$ is generated by translations with the same translation length $\\ell$. We use this to show that $\\mathrm{GR}(\\phi)$ of a monomorphism $\\phi$ of $\\mathcal{H}_n$ is precisely $\\ell$ for all $2\\leq n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-30T22:42:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E36",
      "20F28",
      "20K30"
    ],
    "keywords": [
      "growth rate",
      "houghtons group",
      "endomorphism",
      "unique natural number",
      "discrete points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151200079L"
    }
  }
}