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Abstract.It is shown that if M is a closed orientable irreducible 3-manifold and n is a nonnegative integer, and if HX(M, Zp) has rank > n + 2 for some prime p , then every «-generator subgroup of nx(M) has infinite index in nx(M), and is in fact contained in infinitely many finite-index subgroups of 7tx(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 Zp-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 Zp -homology, and a sufficient condition for an irreducible 3-manifold to be almost sufficiently large. This paper addresses several different problems in the geometric and topological theory of 3-manifolds. In particular, we obtain new results on the problem of estimating the growth rate of the fundamental group of a 3-manifold ( §4); on the problem of estimating numerical invariants of a hyperbolic 3-manifold, such as its volume or its maximal injectivity radius ( §5); and on certain older problems in the topological theory, such as when a 3-manifold A7 is almost sufficiently large, and when nx (M) contains a free subgroup of rank 2 ( §2). All these problems will be seen to be related to a circle of intriguing questions about 3-manifold groups which are treated in § 1.In the following discussion we shall let M denote a closed, orientable 3-manifold with a smooth or piecewise-linear structure. (The results of the paper can be adapted to nonclosed 3-manifolds, but we have concentrated on the closed case as it is the most interesting.) We shall assume that A7 is irreducible: this means that A7 is connected and that every (smooth or PL) 2-sphere in A7 bounds a 3-ball. (According to the prime decomposition theorem for 3-manifolds [Mil], this is not a serious restriction. Furthermore, any hyperbolic 3-manifold is irreducible since it is covered by E3 : see [Mil, proof of Theorem 2]-)In §1 we exploit the idea that the presence of "enough" first homology in the 3-manifold A7 can be helpful in studying 7ix(M).

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