{
  "id": "1110.6453",
  "version": "v1",
  "published": "2011-10-28T20:08:29.000Z",
  "updated": "2011-10-28T20:08:29.000Z",
  "title": "The simple complexity of a Riemann surface",
  "authors": [
    "Aldo-Hilario Cruz-Cota",
    "Teresita Ramirez-Rosas"
  ],
  "comment": "9 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "\\noindent Given a Riemann surface $M$, the \\emph{complexity} of a branched cover of $M$ to the Riemann sphere $S^2$, of degree $d$ and with branching set of cardinality $n \\geq 3$, is defined as $d$ times the hyperbolic area of the complement of its branching set in $S^2$. A branched cover $p \\colon M \\to S^2$ of degree $d$ is \\emph{simple} if the cardinality of the pre-image $p^{-1}(y)$ is at least $d-1$ for all $y \\in S^2$. The \\emph{(simple) complexity} of $M$ is defined as the infimum of the complexities of all (simple) branched covers of $M$ to $S^2$. We prove that if $M$ is a closed, connected, orientable Riemann surface of genus $g \\geq 1$, then: (1) its simple complexity equals $8\\pi g$, and (2) its complexity equals $2\\pi(m_{\\text{min}}+2g-2)$, where $m_{\\text{min}}$ is the minimum total length of a branch datum realizable by a branched cover $p \\colon M \\to S^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-28T20:08:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M12"
    ],
    "keywords": [
      "branched cover",
      "simple complexity equals",
      "minimum total length",
      "branching set",
      "orientable riemann surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6453C"
    }
  }
}