{
  "id": "2302.08184",
  "version": "v1",
  "published": "2023-02-16T10:10:34.000Z",
  "updated": "2023-02-16T10:10:34.000Z",
  "title": "Parabolic isometries of the fine curve graph of the torus",
  "authors": [
    "Pierre-Antoine Guiheneuf",
    "Emmanuel Militon"
  ],
  "categories": [
    "math.DS",
    "math.GR",
    "math.GT"
  ],
  "abstract": "In this article we finish the classification of actions of torus homeomorphisms on the fine curve graph initiated by Bowden, Hensel, Mann, Militon, and Webb in [BHM + 22]. This is made by proving that if $f \\in \\mathrm{Homeo}(\\mathbb{T}^2)$, then $f$ acts elliptically on $C^{\\dagger}(\\mathbb{T} ^2)$ if and only if $f$ has bounded deviation from some $v \\in \\mathbb{Q}^2 \\setminus \\left\\{ 0 \\right\\}$. The proof involves some kind of slow rotation sets for torus homeomorphisms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-16T10:10:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fine curve graph",
      "parabolic isometries",
      "torus homeomorphisms",
      "slow rotation sets",
      "classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}