The ratio of homology rank to hyperbolic volume, II: The role of the Four Color Theorem.

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) π 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 dim H 1 (M ; F p) < 1 5 7. 7 6 3 ⋅ vol (M). If M has one or more cusps, we get a very similar bound assuming that π 1 (M) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus g for g = 2 , ... , 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 1 6 8 instead of 1 5 8 , 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 π 1 (M) in terms of vol M , assuming that either π 1 (M) is 9 -free, or M is closed and π 1 (M) is 5 -free. [ABSTRACT FROM AUTHOR]