{
  "id": "2405.15378",
  "version": "v1",
  "published": "2024-05-24T09:23:57.000Z",
  "updated": "2024-05-24T09:23:57.000Z",
  "title": "Dominating surface-group representations via Fock-Goncharov coordinates",
  "authors": [
    "Pabitra Barman",
    "Subhojoy Gupta"
  ],
  "comment": "39 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $\\rho:\\pi_1(S) \\rightarrow \\mathrm{PSL}_n(\\mathbb{C})$, there exists a positive representation $\\rho_0:\\pi_1(S) \\rightarrow \\mathrm{PSL}_n(\\mathbb{R})$ that dominates $\\rho$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\\mathbb{X}_n= \\mathrm{PSL}_n(\\mathbb{C})/\\mathrm{PSU}(n)$. Moreover, the $\\rho_0$-lengths of peripheral curves remain unchanged. The dominating representation $\\rho_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-24T09:23:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dominating surface-group representations",
      "fock-goncharov coordinates",
      "peripheral curves remain",
      "translation length spectrum",
      "hilbert length spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}