Explicit rank bounds for cyclic covers

Let $M$ be a closed, orientable hyperbolic 3-manifold and $\phi$ a homomorphism of its fundamental group onto $\mathbb{Z}$ that is not induced by a fibration over the circle. For each natural number $n$ we give an explicit lower bound, linear in $n$, on rank of the fundamental group of the cover of $M$ corresponding to $\phi^{-1}(n\mathbb{Z})$. The key new ingredient is the following result: for such a manifold $M$ and a connected, two-sided incompressible surface of genus $g$ in $M$ that is not a fiber or semi-fiber, a reduced homotopy in $(M,S)$ has length at most $14g-12$.