Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds

It is shown that if $M$ is a closed orientable irreducible $3$-manifold and $n$ is a nonnegative integer, and if ${H_1}(M,{\mathbb{Z}_p})$ has rank $\geq n + 2$ for some prime $p$, then every $n$-generator subgroup of ${\pi _1}\,(M)$ has infinite index in ${\pi _1}\,(M)$, and is in fact contained in infinitely many finite-index subgroups of ${\pi _1}\,(M)$. This result is used to estimate the growth rates of the fundamental group of a $3$-manifold in terms of the rank of the ${\mathbb{Z}_p}$-homology. In particular it is used to show that the fundamental group of any closed hyperbolic $3$-manifold has uniformly exponential growth, in the sense that there is a lower bound for the exponential growth rate that depends only on the manifold and not on the choice of a finite generating set. The result also gives volume estimates for hyperbolic $3$-manifolds with enough ${\mathbb{Z}_p}$-homology, and a sufficient condition for an irreducible $3$-manifold to be almost sufficiently large.