{
  "id": "math/0211139",
  "version": "v1",
  "published": "2002-11-08T00:14:25.000Z",
  "updated": "2002-11-08T00:14:25.000Z",
  "title": "Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups",
  "authors": [
    "Elmas Irmak"
  ],
  "comment": "34 pages, 17 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $S$ be a closed, connected, orientable surface of genus at least 3, $\\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $\\lambda: \\mathcal{C}(S) \\to \\mathcal{C}(S)$, preserves nondisjointness (i.e. if $\\alpha$ and $\\beta$ are two vertices in $\\mathcal{C}(S)$ and $i(\\alpha, \\beta) \\neq 0$, then $i(\\lambda(\\alpha), \\lambda(\\beta)) \\neq 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mod_S^*$ and $f: K \\to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism of $S$ and $f$ has a unique extension to an automorphism of $Mod_S^*$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-08T00:14:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M99",
      "20F38"
    ],
    "keywords": [
      "superinjective simplicial maps",
      "injective homomorphism",
      "finite index subgroup",
      "extended mapping class group",
      "unique extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11139I"
    }
  }
}