{
  "id": "2403.06058",
  "version": "v1",
  "published": "2024-03-10T01:27:21.000Z",
  "updated": "2024-03-10T01:27:21.000Z",
  "title": "Volume and topology of bounded and closed hyperbolic 3-manifolds, II",
  "authors": [
    "Jason DeBlois",
    "Peter B. Shalen"
  ],
  "comment": "48 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $N$ be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus $2$. If $N$ has Heegaard genus at least $5$, then its volume is greater than $2V_{\\rm oct}$, where $V_{\\rm oct}=3.66\\ldots$ denotes the volume of a regular ideal hyperbolic octahedron in $\\mathbb{H}^3$. This improves the lower bound given in our earlier paper ``Volume and topology of bounded and closed hyperbolic $3$-manifolds.'' One ingredient in the improved bound is that in a crucial case, instead of using a single ``muffin'' in $N$ in the sense of Kojima and Miyamoto, we use two disjoint muffins. By combining the result about manifolds with geodesic boundary with the $\\log(2k-1)$ theorem and results due to Agol-Culler-Shalen and Shalen-Wagreich, we show that if $M$ is a closed, orientable hyperbolic $3$-manifold with $\\mathop{\\rm vol} M\\le V_{\\rm oct}/2$, then $\\dim H_1(M;\\mathbb{F}_2)\\le4$. We also provide new lower bounds for the volumes of closed hyperbolic $3$-manifolds whose cohomology ring over $\\mathbb{F}_2$ satisfies certain restrictions; these improve results that were proved in ``Volume and topology$\\ldots$.''",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-10T01:27:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K32"
    ],
    "keywords": [
      "closed hyperbolic",
      "lower bound",
      "regular ideal hyperbolic octahedron",
      "orientable hyperbolic",
      "connected totally geodesic surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}