{
  "id": "2410.15488",
  "version": "v1",
  "published": "2024-10-20T20:04:39.000Z",
  "updated": "2024-10-20T20:04:39.000Z",
  "title": "On the topology of manifolds with nonnegative Ricci curvature and linear volume growth",
  "authors": [
    "Dimitri Navarro",
    "Jiayin Pan",
    "Xingyu Zhu"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\\mathbb{Z}^k$ of finite index, where $0\\le k\\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-20T20:04:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear volume growth",
      "nonnegative ricci curvature",
      "fundamental group",
      "equivariant asymptotic geometry",
      "plane/halfplane rigidity result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}