{
  "id": "0802.2731",
  "version": "v5",
  "published": "2008-02-19T23:02:26.000Z",
  "updated": "2011-02-12T13:28:15.000Z",
  "title": "Enumerating Palindromes and Primitives in Rank Two Free Groups",
  "authors": [
    "Jane Gilman",
    "Linda Keen"
  ],
  "comment": "Final revisions, to appear J Algebra",
  "categories": [
    "math.GR",
    "math.CV",
    "math.GT",
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\\sl primitive} if it, along with another group element, generates the group. It is a {\\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only a representative for the conjugacy class. Here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes. We denote these words by $E_{p/q}$ where $p/q$ is rational number expressed in lowest terms. We prove that $E_{p/q}$ is a palindrome if $pq$ is even and the unique product of two unique palindromes if $pq$ is odd. We prove that the pairs $(E_{p/q},E_{r/s})$ generate the group when $|ps-rq|=1$. This improves the previously known result that held only for $pq$ and $rs$ both even. The derivation of the enumeration scheme also gives a new proof of the known results about primitives.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-02-12T13:28:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20H10",
      "20F10",
      "32G15",
      "30F40",
      "30F60",
      "32G15",
      "11A55",
      "30B70",
      "30F10"
    ],
    "keywords": [
      "free group",
      "enumerating palindromes",
      "unique palindrome",
      "conjugacy class",
      "iteration scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.2731G"
    }
  }
}