The ratio of homology rank to hyperbolic volume, I

Rosemary K. Guzman, Peter B. Shalen

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $\text{dim}{H}_1(M;{\mathbf{F}}_p)<168.602\cdot \text{vol}M$. There are slightly strong…