{
  "id": "2401.02560",
  "version": "v1",
  "published": "2024-01-04T22:17:49.000Z",
  "updated": "2024-01-04T22:17:49.000Z",
  "title": "Asymptotic dimension and geometric decompositions in dimensions 3 and 4",
  "authors": [
    "H. Contreras Peruyero",
    "P. Suárez-Serrato"
  ],
  "comment": "22 pages, 2 images",
  "categories": [
    "math.GT",
    "math.DG",
    "math.GR"
  ],
  "abstract": "We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most 3. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most 3. We thus obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-04T22:17:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic dimension",
      "dimensional alexandrov spaces",
      "admit geometric decompositions",
      "proof method yields",
      "novikov conjecture holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}