2022/06/30 by Rosemary K. Guzman, Peter B. Shalen, Guzman, Rosemary K. +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2207.00040
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) π1(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 H1(M;Fp) < 157.763 ⋅ vol(M). If M has one or more cusps, we get a very similar bound assuming that π1(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 π1(M) in terms of vol M, assuming that either π1(M) is 9-free, or M is closed and π1(M) is 5-free.