Growth rates, 𝑍_{𝑝}-homology, and volumes of hyperbolic 3-manifolds

Shalen, Peter B.; Wagreich, Philip

doi:10.1090/S0002-9947-1992-1156298-8

Growth rates, $Z_ p$-homology, and volumes of hyperbolic $3$-manifolds
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Trans. Amer. Math. Soc. 331 (1992), 895-917 Request permission

Abstract:

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.

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