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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Peter B. Shalen and Philip Wagreich
Journal: Trans. Amer. Math. Soc. 331 (1992), 895-917
MSC: Primary 57M05; Secondary 20F05, 57M07, 57N10
MathSciNet review: 1156298
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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 $ {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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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1156298-8
PII: S 0002-9947(1992)1156298-8
Article copyright: © Copyright 1992 American Mathematical Society