{
  "id": "0802.0584",
  "version": "v1",
  "published": "2008-02-05T10:37:13.000Z",
  "updated": "2008-02-05T10:37:13.000Z",
  "title": "On several problems about automorphisms of the free group of rank two",
  "authors": [
    "Donghi Lee"
  ],
  "comment": "30 pages",
  "journal": "J. Algebra, vol.321 (2009), pp.167-193",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called potentially positive if $\\phi(u)$ is positive for some automorphism $\\phi$ of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of $F_2$. Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $\\phi(u)$ and $\\phi(v)$ is bounded away from 0 and from $\\infty$ for every automorphism $\\phi$ of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of $F_2$. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in [1] in the affirmative for the case of $F_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-02-05T10:37:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E36",
      "20F05",
      "20F10",
      "20F28"
    ],
    "keywords": [
      "free group",
      "automorphism",
      "boundedly translation equivalent",
      "settles problem f1b",
      "algorithmic problems"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.0584L"
    }
  }
}