The ratio of homology rank to hyperbolic volume, II

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) 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 ) < 157.763•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 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 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.

Euler characteristics, free subgroups, and lengths of loops in hyperbolic 3-manifolds

“…Then χ(G) ≤ k − 2. This theorem will be used in a forthcoming paper with Rosemary Guzman to obtain a result that gives a new bound on the ratio of the dimension of the mod 2 homology of a closed, orientable 3-manifold to the volume; this result is strictly stronger than the one established in [9], and is essentially stronger than the one established in [10].…”