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Under mild topological restrictions, we obtain new linear upper bounds forthe dimension of the mod $p$ homology (for any prime $p$) of a finite-volumeorientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprisingfeature of the arguments in the paper is that they require an application ofthe Four Color Theorem.  If $M$ is closed, and either (a) $\pi_1(M)$ has no subgroup isomorphic to thefundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or(b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surfaceof 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 boundassuming that $\pi_1(M)$ has no subgroup isomorphic to the fundamental group ofa closed, orientable surface of genus $g$ for $g = 2, \dots,8$. These resultsshould be compared with those of our previous paper $The\ ratio\ of\ homology\rank\ to\ hyperbolic\ volume,\ I$, in which we obtained a bound with acoefficient in the range of $168$ instead of $158$, without a restriction onsurface subgroups or incompressible surfaces. In a future paper, using a muchmore involved argument, we expect to obtain bounds close to those given by thepresent paper without such a restriction.  The arguments also give new linear upper bounds (with constant terms) for therank 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.

The ratio of homology rank to hyperbolic volume, II