Explicit rank bounds for cyclic covers

Abstract

For a closed, orientable hyperbolic $3$–manifold $M$ and an onto homomorphism $\varphi :{\pi }_1\left(M\right)\to \mathbb{Z}$ that is not induced by a fibration $M\to S^1$, we bound the ranks of the subgroups ${\varphi }^{-1}\left(n\mathbb{Z}\right)$ for $n\in \mathbb{N}$, below, linearly in $n$. The key new ingredient is the following result: if $M$ is a closed, orientable hyperbolic $3$–manifold and $S$ is a connected, two-sided incompressible surface of genus $g$ that is not a fiber or semifiber, then a reduced homotopy in $\left(M,S\right)$ has length at most $14g-12$.