{
  "id": "1809.10220",
  "version": "v1",
  "published": "2018-09-26T20:33:59.000Z",
  "updated": "2018-09-26T20:33:59.000Z",
  "title": "Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups",
  "authors": [
    "Jiayin Pan"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "This is a continuation of the author's work arXiv:1710.05498. We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\\widetilde{M}$, the Riemannian universal cover of $M$. We show that if any tangent cone of $\\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $\\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, this result confirms the Milnor conjecture for any manifold whose universal cover has Euclidean volume growth of constant at least $1-\\epsilon(n)$, where $\\epsilon(n)>0$ is some universal constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-26T20:33:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative ricci curvature",
      "fundamental group",
      "euclidean volume growth",
      "riemannian universal cover",
      "additional stability condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}