  We establish new strong lower bounds on the (subnormal) subgroup growth of a
large class of groups. This includes the fundamental groups of all
finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic
groups. The lower bound is nearly exponential, which should be compared with
the fastest possible subgroup growth of any finitely generated group. This is
achieved by free non-abelian groups and is slightly faster than exponential. As
a consequence, we obtain good estimates on the number of covering spaces of a
hyperbolic 3-manifold with given covering degree. We also obtain slightly
weaker information on the number of covering spaces of closed 4-manifolds with
non-positive Euler characteristic. The results on subgroup growth follow from a
new theorem which places lower bounds on the rank of the first homology (with
mod p coefficients) of certain subgroups of a group. This is proved using a
topological argument.
