{
  "id": "1905.13366",
  "version": "v1",
  "published": "2019-05-31T00:59:48.000Z",
  "updated": "2019-05-31T00:59:48.000Z",
  "title": "Local estimate of fundamental groups",
  "authors": [
    "Guoyi Xu"
  ],
  "comment": "to appear in Adv. Math",
  "categories": [
    "math.DG",
    "math.AT"
  ],
  "abstract": "For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group $\\pi_1(M^n)$ can be generated by $C(n)$ generators. Inspired by their work, we give a quantitative proof of the above theorem and show that $C(n)\\leq n^{n^{20n}} $. Our main tools are quantitative Cheeger-Colding's almost splitting theory, and the squeeze lemma for covering groups between two Riemannian manifolds with nonnegative Ricci curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-31T00:59:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fundamental group",
      "local estimate",
      "nonnegative ricci curvature",
      "dim riemannian manifold",
      "main tools"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}