{
  "id": "2012.07417",
  "version": "v2",
  "published": "2020-12-14T11:09:38.000Z",
  "updated": "2021-04-27T04:24:35.000Z",
  "title": "The fundamental inequality for cocompact Fuchsian groups",
  "authors": [
    "Petr Kosenko",
    "Giulio Tiozzo"
  ],
  "comment": "20 pages, 5 figures. Main result (Theorem 1) strengthened; results on Coxeter groups (Theorem 2) and Hausdorff dimension (Corollary 3) added. Introduction expanded",
  "categories": [
    "math.DS",
    "math.GT",
    "math.PR"
  ],
  "abstract": "We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff dimension of the hitting measure is strictly less than 1. A similar statement is proven for Coxeter groups. Along the way, we prove for cocompact Fuchsian groups a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-04-27T04:24:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "20F67",
      "30F35",
      "60J50"
    ],
    "keywords": [
      "cocompact fuchsian group",
      "fundamental inequality",
      "hitting measure",
      "translations joining opposite sides",
      "symmetric hyperbolic polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}