{
  "id": "1812.00731",
  "version": "v1",
  "published": "2018-12-03T13:40:00.000Z",
  "updated": "2018-12-03T13:40:00.000Z",
  "title": "Projections of Poisson cut-outs in the Heisenberg group and the visual $3$-sphere",
  "authors": [
    "Laurent Dufloux",
    "Ville Suomala"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR",
    "math.CA",
    "math.DG",
    "math.DS"
  ],
  "abstract": "We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Kor\\'anyi metric, we show that the Hausdorff dimension of the vertical projection $\\pi(E)$ (projection along the center of the Heisenberg group) almost surely equals $\\min\\{2,dim_H(E)\\}$ and that $\\pi(E)$ has non-empty interior if $dim_H(E)>2$. As a corollary, this allows us to determine the Hausdorff dimension of $E$ with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension $dim_H (E)$. We also study projections in the one-point compactification of the Heisenberg group, that is, the $3$-sphere $S^3$ endowed with the visual metric $d$ obtained by identifying $S^3$ with the boundary of the complex hyperbolic plane. In $S^3$, we prove a projection result that holds simultaneously for all radial projections (projections along so called \"chains\"). This shows that the Poisson cut-outs in $S^3$ satisfy a strong version of the Marstrand's projection theorem, without any exceptional directions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-03T13:40:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "28A80",
      "37D35",
      "37C45",
      "53C17"
    ],
    "keywords": [
      "heisenberg group",
      "first heisenbeg group",
      "poisson cut-out sets",
      "marstrands projection theorem",
      "study projectional properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}