{
  "id": "1909.03097",
  "version": "v1",
  "published": "2019-09-06T19:14:19.000Z",
  "updated": "2019-09-06T19:14:19.000Z",
  "title": "Train Tracks, Orbigraphs and CAT(0) Free-by-cyclic Groups",
  "authors": [
    "Rylee Alanza Lyman"
  ],
  "comment": "39 pages, 4 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Gersten gave an example of a polynomially-growing automorphism of $F_3$ whose mapping torus $F_3 \\rtimes \\mathbb{Z}$ cannot act properly by semi-simple isometries on a CAT(0) metric space. By contrast, we show that if $\\Phi$ is a polynomially-growing automorphism belonging to one of several related groups, there exists $k > 0$ such that the mapping torus of $\\Phi^k$ acts properly and cocompactly on a CAT(0) metric space. This $k$ can often be bounded uniformly. Our results apply to automorphisms of a free product of $n$ copies of a finite group $A$, as well as to palindromic and symmetric automorphisms of a free group of finite rank. Of independent interest, a key tool in our proof is the construction of relative train track maps on orbigraphs, certain graphs of groups thought of as orbi-spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-06T19:14:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "20E08",
      "57M07"
    ],
    "keywords": [
      "free-by-cyclic groups",
      "orbigraphs",
      "metric space",
      "polynomially-growing automorphism",
      "relative train track maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}