{
  "id": "1710.09286",
  "version": "v1",
  "published": "2017-10-24T06:26:35.000Z",
  "updated": "2017-10-24T06:26:35.000Z",
  "title": "Bordered surfaces in the 3-sphere with maximum symmetry",
  "authors": [
    "Chao Wang",
    "Shicheng Wang",
    "Yimu Zhang",
    "Bruno Zimmermann"
  ],
  "comment": "20 pages, to appear in J. Pure Appl. Algebra. arXiv admin note: text overlap with arXiv:1510.00822",
  "categories": [
    "math.GT"
  ],
  "abstract": "We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \\Sigma)$, where $\\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus $g>1$ the maximum order of such a $G$ for all embeddings of a surface of genus $g$, and classified the corresponding embeddings. In the present paper we obtain analogous results for the case of bordered surfaces $\\Sigma$ (i.e. with non-empty boundary, orientable or not). Now the genus $g$ gets replaced by the algebraic genus $\\alpha$ of $\\Sigma$ (the rank of its free fundamental group); for each $\\alpha > 1$ we determine the maximum order $m_\\alpha$ of an action of $G$, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into $S^3$. For example, the maximal possibility $12(\\alpha - 1)$ is obtained for the finitely many values $\\alpha = 2, 3, 4, 5, 9, 11, 25, 97, 121$ and $241$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-24T06:26:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M60",
      "57S25"
    ],
    "keywords": [
      "bordered surfaces",
      "maximum symmetry",
      "maximum order",
      "free fundamental group",
      "embeddings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}