{
  "id": "1711.01222",
  "version": "v1",
  "published": "2017-11-03T16:12:04.000Z",
  "updated": "2017-11-03T16:12:04.000Z",
  "title": "Rigidity at infinity for lattices in rank-one Lie groups",
  "authors": [
    "Alessio Savini"
  ],
  "comment": "18 pages",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let $\\Gamma$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\\geq2 $. We introduce the notion of volume for a representation $\\rho:\\Gamma \\rightarrow PU(m,1)$ where $m \\geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $\\rho_n:\\Gamma \\rightarrow PU(m,1)$ such that $\\lim_{n \\to \\infty} \\text{Vol}(\\rho_n) =\\text{Vol}(M)$, then there must exist a sequence of elements $g_n \\in PU(m,1)$ such that the representations $g_n \\circ \\rho_n \\circ g_n^{-1}$ converge to a reducible representation $\\rho_\\infty$ which preserves a totally geodesic copy of $\\mathbb{H}^p_\\mathbb{C}$ and whose $\\mathbb{H}^p_\\mathbb{C}$-component is conjugated to the standard lattice embedding $i:\\Gamma \\rightarrow PU(p,1) < PU(m,1)$. Additionally, we show that the same definitions and results can be adapted when $\\Gamma$ is a non-uniform lattice of $PSp(p,1)$ without torsion and for representations $\\rho:\\Gamma \\rightarrow PSp(m,1)$, still mantaining the hypothesis $m \\geq p \\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-03T16:12:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C24",
      "53C35",
      "22E40"
    ],
    "keywords": [
      "rank-one lie groups",
      "non-uniform lattice",
      "mostow-prasad rigidity theorem",
      "standard lattice",
      "totally geodesic copy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}