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 $\mathrm{\text{dim}}{H}_1(M;F_p)<157.763\cdot \mathrm{\text{vol}}(M)$. 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 this 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 $\mathrm{\text{vol}}M$, assuming that either ${\pi }_1(M)$ is $9$-free, or $M$ is closed and ${\pi }_1(M)$ is $5$-free.

在温和的拓扑限制下，我们获得了 mod 维数的新线性上限$p$同源性（对于任何素数$p$) 的有限体积可定向双曲线$3$-歧管$\mathrm{中号}$就其体积而言。论文中的论证的一个令人惊讶的特点是它们需要应用四色定理。如果$\mathrm{中号}$是封闭的，并且 (a)${\pi }_1（\mathrm{中号}）$没有与属的封闭可定向曲面的基本群同构的子群$2$,$3$或者$4$，或（b）$p=2$， 和$\mathrm{中号}$不包含属的（嵌入的、两侧的）不可压缩表面$2$,$3$或者$4$， 然后$\mathrm{\text{暗淡}}{H}_1（\mathrm{中号};F_p）<157。763\cdot \mathrm{\text{卷}}（\mathrm{中号}）$。如果$\mathrm{中号}$有一个或多个尖点，我们得到一个非常相似的界限，假设${\pi }_1（\mathrm{中号}）$没有与属的封闭可定向曲面的基本群同构的子群$G$为了$G=2,\dots \dots ,8$。这些结果应该与我们之前的论文“同源等级与双曲体积之比，I”进行比较，其中我们获得了系数范围为$168$代替$158$，对曲面子群或不可压缩曲面没有限制。在未来的论文中，使用更复杂的论证，我们期望获得接近本文给出的界限，而没有这样的限制。这些参数还给出了新的线性上限（带有常数项）${\pi }_1（\mathrm{中号}）$按照$\mathrm{\text{卷}}\mathrm{中号}$，假设${\pi }_1（\mathrm{中号}）$是$9$-免费，或$\mathrm{中号}$已关闭并且${\pi }_1（\mathrm{中号}）$是$5$-自由的。