{
  "id": "1501.06723",
  "version": "v1",
  "published": "2015-01-27T10:17:14.000Z",
  "updated": "2015-01-27T10:17:14.000Z",
  "title": "Fixed subgroups are compressed in surface groups",
  "authors": [
    "Qiang Zhang",
    "Enric Ventura",
    "Jianchun Wu"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "For a compact surface $\\Sigma$ (orientable or not, and with boundary or not) we show that the fixed subgroup, $\\operatorname{Fix} B$, of any family $B$ of endomorphisms of $\\pi_1(\\Sigma)$ is compressed in $\\pi_1(\\Sigma)$ i.e., $\\operatorname{rk}((\\operatorname{Fix} B)\\cap H)\\leq \\operatorname{rk}(H)$ for any subgroup $\\operatorname{Fix} B \\leq H \\leq \\pi_1(\\Sigma)$. On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, $G$, of finitely many free and surface groups, and give a characterization of when $G$ satisfies that $\\operatorname{rk}(\\operatorname{Fix} \\phi) \\leq \\operatorname{rk}(G)$ for every $\\phi \\in Aut(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-27T10:17:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F34"
    ],
    "keywords": [
      "surface groups",
      "fixed subgroup",
      "compact surface",
      "partial positive solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150106723Z"
    }
  }
}