{
  "id": "1306.6025",
  "version": "v3",
  "published": "2013-06-25T16:37:23.000Z",
  "updated": "2015-06-30T20:35:43.000Z",
  "title": "Coxeter groups, hyperbolic cubes, and acute triangulations",
  "authors": [
    "Sang-hyun Kim",
    "Genevieve S. Walsh"
  ],
  "comment": "27 pages, 9 figures. Accepted for publication by the Journal of Topology",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called \"flag no-square\". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-01T20:01:24.000Z",
      "abstract": "Let C(L) be the right-angled Coxeter group defined by an abstract triangulation L of S^2. We show that C(L) is isomorphic to a hyperbolic right-angled reflection group if and only if L can be realized as an acute triangulation. A corollary is that an abstract triangulation of S^2 can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called \"flag no-square\". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.",
      "comment": "27 pages, 9 figures. References corrected in this version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-06-30T20:35:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M07",
      "20F65",
      "37F20"
    ],
    "keywords": [
      "hyperbolic cubes",
      "abstract triangulation",
      "hyperbolic right-angled reflection group",
      "right-angled coxeter group",
      "planar surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6025K"
    }
  }
}