{
  "id": "0902.0202",
  "version": "v6",
  "published": "2009-02-02T05:42:39.000Z",
  "updated": "2010-02-19T00:24:13.000Z",
  "title": "Counting elements and geodesics in Thompson's group $F$",
  "authors": [
    "Murray Elder",
    "Eric Fusy",
    "Andrew Rechnitzer"
  ],
  "comment": "27 pages, 9 figures, 3 tables",
  "doi": "10.1016/j.jalgebra.2010.02.035",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We present two quite different algorithms to compute the number of elements in the sphere of radius $n$ of Thompson's group $F$ with standard generating set. The first of these requires exponential time and polynomial space, but additionally computes the number of geodesics and is generalisable to many other groups. The second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius $n$ with $n \\leq 1500$. Using the resulting series data we find that the growth rate of the group is bounded above by $2.62167...$. This is very close to Guba's lower bound of $\\tfrac{3+\\sqrt{5}}{2}$ \\cite{Guba2004}. Indeed, numerical analysis of the series data strongly suggests that the growth rate of the group is exactly $\\tfrac{3+\\sqrt{5}}{2}$.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2010-02-19T00:24:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65"
    ],
    "keywords": [
      "thompsons group",
      "counting elements",
      "growth rate",
      "gubas lower bound",
      "standard generating set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0202E"
    }
  }
}