Explicit rank bounds for cyclic covers

For a closed, orientable hyperbolic 3-manifold M and an onto homomorphism phi: pi(1) (M) -> Z that is not induced by a fibration M -> S-1, we bound the ranks of the subgroups phi(-1) (nZ) for n epsilon 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 (M, S) has length at most 14g - 12.