The ratio of homology rank to hyperbolic volume

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.