{
  "id": "2207.00040",
  "version": "v1",
  "published": "2022-06-30T18:16:18.000Z",
  "updated": "2022-06-30T18:16:18.000Z",
  "title": "The ratio of homology rank to hyperbolic volume, II",
  "authors": [
    "Rosemary K. Guzman",
    "Peter B. Shalen"
  ],
  "comment": "37 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $\\pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $\\text{dim}\\, H_1(M;F_p) < 157.763 \\cdot \\text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $\\pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, \\dots,8$. These results should be compared with those of our previous paper $The\\ ratio\\ of\\ homology\\ rank\\ to\\ hyperbolic\\ volume,\\ I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $\\pi_1(M)$ in terms of $\\text{vol}\\,M$, assuming that either $\\pi_1(M)$ is $9$-free, or $M$ is closed and $\\pi_1(M)$ is $5$-free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-30T18:16:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K32"
    ],
    "keywords": [
      "hyperbolic volume",
      "homology rank",
      "linear upper bounds",
      "fundamental group",
      "subgroup isomorphic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}