{
  "id": "1306.4711",
  "version": "v3",
  "published": "2013-06-19T22:30:42.000Z",
  "updated": "2015-07-28T15:03:53.000Z",
  "title": "Subgroup decomposition in Out(F_n), Part IV: Relatively irreducible subgroups",
  "authors": [
    "Michael Handel",
    "Lee Mosher"
  ],
  "comment": "The latest version includes better structured proofs, and stronger statements of theorems for later application. 32 pages. Cross references to other parts this series are to the June 2013 versions. All other parts of this series, including the research announcement, are found on this arXiv",
  "categories": [
    "math.GR"
  ],
  "abstract": "This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $Out(F_n)$. In this paper we develop general ping-pong techniques for the action of $Out(F_n)$ on the space of lines of $F_n$. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup $\\mathcal H$ of $Out(F_n)$ that acts trivially on homology with $\\mathbb{Z}/3$ coefficients, and for any free factor system $\\mathcal F$ that does not consist of (the conjugacy classes of) a complementary pair of free factors of $F_n$ nor of a rank $n-1$ free factor, if $\\mathcal H$ is fully irreducible relative to $\\mathcal F$ then $\\mathcal H$ has an element that is fully irreducible relative to $\\mathcal F$. We also prove Theorem J which, under the additional hypothesis that $\\mathcal H$ is geometric relative to $\\mathcal F$, describes a strong relation between $\\mathcal H$ and a mapping class group of a surface.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-21T15:53:25.000Z",
      "abstract": "This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of Out(F_n). In this paper we develop general ping-pong techniques for the action of Out(F_n) on the space of lines of F_n. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup H of Out(F_n) that acts trivially on homology with Z/3 coefficients, and for any free factor system F that does not consist of (the conjugacy classes of) a complementary pair of free factors of F_n nor of a rank n-1 free factor, if H is fully irreducible relative to F then H has an element that is fully irreducible relative to F. We also prove Theorem J which, under the additional hypothesis that H is geometric relative to F, describes a strong relation between H and a mapping class group of a surface.",
      "comment": "27 pages. Cross references to other parts this series are to the June 2013 versions. All other parts, including the research announcement, are found on this arXiv",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-07-28T15:03:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "57M07"
    ],
    "keywords": [
      "relatively irreducible subgroups",
      "subgroup decomposition",
      "research announcement",
      "general ping-pong techniques",
      "special case theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.4711H"
    }
  }
}