{
  "id": "2311.03709",
  "version": "v1",
  "published": "2023-11-07T04:22:38.000Z",
  "updated": "2023-11-07T04:22:38.000Z",
  "title": "Geodesic Envelopes in the Thurston Metric on Teichmuller space",
  "authors": [
    "Assaf Bar-Natan"
  ],
  "comment": "26 pages, 7.5 figures. Comments and criticism welcome. This is a no-frills version of my PhD thesis :)",
  "categories": [
    "math.GT"
  ],
  "abstract": "The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y) = \\frac12 log\\sup_{\\alpha} \\frac{l_{\\alpha}(Y)}{l_{\\alpha}(X)}$. This metric is geodesic, but geodesics are far from unique. In this thesis, we show that in the once-punctured torus, and in the four-times punctured sphere, geodesics stay a uniformly-bounded distance from each other. In other words, we show that the \\textbf{width} of the geodesic envelope, E(X,Y) between any pair of points $X,Y \\in T(S)$ (where $S = S_{1,1}$ or $S = S_{0,4}$) is bounded uniformly. To do this, we first identify extremal geodesics in Env(X,Y), and show that these correspond to stretch vectors, proving a conjecture of Huang, Ohshika and Papadopoulos. We then compute Fenchel-Nielsen twisting along these paths, and use these computations, along with estimates on earthquake path lengths, to prove the main theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-07T04:22:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "teichmuller space",
      "thurston metric",
      "geodesic envelope",
      "first identify extremal geodesics",
      "earthquake path lengths"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}