{
  "id": "1609.04118",
  "version": "v1",
  "published": "2016-09-14T03:13:23.000Z",
  "updated": "2016-09-14T03:13:23.000Z",
  "title": "Divergent trajectories under diagonal geodesic flow and splitting of discrete subgroups of $\\mathrm{SO}(n,1) \\times \\mathrm{SO}(n,1)$",
  "authors": [
    "Lei Yang"
  ],
  "comment": "6 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $H = \\mathrm{SO}(n,1)$ and $A = \\{a(t): t \\in \\mathbb{R}\\}$ be a maximal $\\mathbb{R}$-split Cartan subgroup of $H$. Let $\\Gamma \\subset H \\times H$ be a nonuniform lattice in $H \\times H$ and $X_{\\Gamma} : = H \\times H/ \\Gamma$. Let $A_2 : = \\{ a_2(t):=a(t) \\times a(t) : t \\in \\mathbb{R}\\} \\subset A\\times A$ on $X_{\\Gamma}$ and $\\mathcal{D}_{\\Gamma}\\subset X_{\\Gamma}$ denote the collection of points $x \\in X_{\\Gamma}$ such that $a_2(t)x$ diverges as $t \\rightarrow +\\infty$. In this note, we will show that if the Hausdorff dimension of $\\mathcal{D}_{\\Gamma}$ is greater than $\\dim (H\\times H) - 2(n-1)$, then $\\Gamma $ is essentially split, namely, it contains a subgroup of finite index of form $\\Gamma_1 \\times \\Gamma_2 $, where $\\Gamma_1$ and $\\Gamma_2$ are both lattices in $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-14T03:13:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "22E40"
    ],
    "keywords": [
      "diagonal geodesic flow",
      "discrete subgroups",
      "divergent trajectories",
      "split cartan subgroup",
      "nonuniform lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}