{
  "id": "2207.12919",
  "version": "v1",
  "published": "2022-07-26T14:15:25.000Z",
  "updated": "2022-07-26T14:15:25.000Z",
  "title": "On second eigenvalues of closed hyperbolic surfaces for large genus",
  "authors": [
    "Yuxin He",
    "Yunhui Wu"
  ],
  "comment": "37 pages, 6 figures, comments are welcome",
  "categories": [
    "math.GT",
    "math.DG",
    "math.SP"
  ],
  "abstract": "In this article the second eigenvalues of closed hyperbolic surfaces for large genus have been studied. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\\geq 3)$, the second eigenvalue $\\lambda_2(X_g)$ of $X_g$ is greater than $\\frac{\\mathcal{L}_2(X_g)}{g^2}$ and less than $\\mathcal{L}_2(X_g)$ up to uniform positive constants multiplications; moreover these two bounds are optimal as $g\\to \\infty$. Where $\\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also study the quantity $\\frac{\\lambda_2(X_g)}{\\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\\to \\infty$, a generic hyperbolic surface $X_g$ has $\\frac{\\lambda_2(X_g)}{\\mathcal{L}_2(X_g)}$ uniformly comparable to $\\frac{1}{\\ln(g)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T14:15:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed hyperbolic surface",
      "second eigenvalue",
      "large genus",
      "generic hyperbolic surface",
      "random hyperbolic surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}