{
  "id": "2502.03259",
  "version": "v1",
  "published": "2025-02-05T15:16:36.000Z",
  "updated": "2025-02-05T15:16:36.000Z",
  "title": "Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds",
  "authors": [
    "Hongzhi Huang",
    "Xian-Tao Huang"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $\\pi_1(M)$ is finitely generated. This result confirms Pan-Rong's conjecture \\cite{PR18} for dimension $n = 4$. Additionally, we prove that there exists a universal constant $C>0$ such that $\\pi_1(M)$ contains an abelian subgroup of index $\\le C$. More specifically, if $\\pi_1(M)$ is infinite, then $\\pi_1(M)$ is a crystallographic group of rank $\\le 3$. If $\\pi_1(M)$ is finite, then $\\pi_1(M)$ is isomorphic to a quotient of the fundamental group of a spherical 3-manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T15:16:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean volume growth",
      "nonnegative ricci curvature",
      "fundamental group",
      "result confirms pan-rongs conjecture",
      "open manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}