{
  "id": "1304.7979",
  "version": "v3",
  "published": "2013-04-30T13:02:07.000Z",
  "updated": "2014-10-23T15:00:13.000Z",
  "title": "Growth of Primitive Elements in Free Groups",
  "authors": [
    "Doron Puder",
    "Conan Wu"
  ],
  "comment": "20 pages, 2 figures. A few minor improvements of the introduction of ideas",
  "journal": "J. London Math. Soc. (2014) 90 (1): 89-104",
  "doi": "10.1112/jlms/jdu009",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "In the free group $F_k$, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length $N$ contains one of the letters exactly once asymptotically almost surely (as $N \\to \\infty$). This also solves a question from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number of primitive words of length $N$ in $F_k$. We show that for $k \\ge 3$, the exponential growth rate of $p_{k,N}$ is $2k-3$. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-10T08:30:41.000Z",
      "abstract": "Let p_{k,N} be the number of primitive words of length N in the free group F_k, k \\ge 3. We show that the exponential growth rate of p_{k,N} is 2k-3, answering a question from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Moreover, we show that a generic conjugacy class of primitive elements has a representative containing some letter exactly once. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.",
      "comment": "17 pages, 2 figures. Affiliations and emails added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-23T15:00:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E05",
      "05A16"
    ],
    "keywords": [
      "free group",
      "primitive elements",
      "combinatorial group theory",
      "proper free factor",
      "exact growth rate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.7979P"
    }
  }
}