Growth rates, -homology, and volumes of hyperbolic -manifolds

Authors:
Peter B. Shalen and Philip Wagreich

Journal:
Trans. Amer. Math. Soc. **331** (1992), 895-917

MSC:
Primary 57M05; Secondary 20F05, 57M07, 57N10

DOI:
https://doi.org/10.1090/S0002-9947-1992-1156298-8

MathSciNet review:
1156298

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Abstract: It is shown that if is a closed orientable irreducible -manifold and is a nonnegative integer, and if has rank for some prime , then every -generator subgroup of has infinite index in , and is in fact contained in infinitely many finite-index subgroups of . This result is used to estimate the growth rates of the fundamental group of a -manifold in terms of the rank of the -homology. In particular it is used to show that the fundamental group of any closed hyperbolic -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 -manifolds with enough -homology, and a sufficient condition for an irreducible -manifold to be almost sufficiently large.

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DOI:
https://doi.org/10.1090/S0002-9947-1992-1156298-8

Article copyright:
© Copyright 1992
American Mathematical Society