{
  "id": "math/0505644",
  "version": "v1",
  "published": "2005-05-30T12:49:42.000Z",
  "updated": "2005-05-30T12:49:42.000Z",
  "title": "Compact Group Actions On Closed Manifolds of Non-positive Curvature",
  "authors": [
    "Bin Xu"
  ],
  "comment": "8 pages",
  "journal": "International Journal of Mathematics Vol. 17, No. 1 (2006) 119-227",
  "categories": [
    "math.DG",
    "math.GT"
  ],
  "abstract": "A. Borel proved that, if a finite group $F$ acts effectively and continuously on a closed aspherical manifold $M$ with centerless fundamental group $\\pi_1(M)$, then a natural homomorphism $\\psi$ from $F$ to the outer automorphism group ${\\rm Out} \\pi_1(M)$ of $\\pi_1(M)$, called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group $G$ acting effectively and smoothly on a particular orientable aspherical manifold $N$ admitting a Riemannian metric $g_0$ of non-positive curvature in case that $\\pi_1(N)$ has a non-trivial center. It turns out that if $G$ attains the maximal dimension equal to the rank of Center $\\pi_1(N)$ and the metric $g_0$ is real analytic, then any element of $G$ defining a diffemorphism homotopic to the identity of $N$ must be contained in the identity component $G^0$ of $G$. Moreover, if the inner automorphism group of $\\pi_1(N)$ is torsion free, then the associated abstract kernel $\\psi: G/G^0\\to {\\rm Out} \\pi_1(N)$ is a monomorphism. The same result holds for the non-orientable $N$'s under certain techical assumptions. Our result is an application of a theorem by Schoen-Yau (Topology, {\\bf 18} (1979), 361-380) on harmonic mappings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-05-30T12:49:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S25",
      "53C43",
      "20F34"
    ],
    "keywords": [
      "compact group actions",
      "non-positive curvature",
      "closed manifolds",
      "associated abstract kernel",
      "extent borels theorem holds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5644X"
    }
  }
}