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Growth rates, \(\mathbb{Z}_p\)-homology, and volumes of hyperbolic 3-manifolds. (English) Zbl 0768.57001

The authors credit A. Lubotzky with inspiring the underlying result of this beautifully written paper: if the first homology \(H_ 1(M)\) (with \(Z_ p\) coefficients, for some prime \(p\)) has rank at least \(n+2\), then every \(n\)-generator subgroup of \(\pi_ 1(M)\) must be contained in infinitely many distinct finite-index subgroups of \(\pi_ 1(M)\) (and hence has infinite index). The authors prove this by fairly elementary but clever arguments, and derive many consequences for longstanding problems in the theory of 3-manifolds. For example, if \(M\) is closed and irreducible (and orientable), and \(H_ 1(M)\) has rank at least \(4\), then \(\pi_ 1(M)\) has a free subgroup of rank \(2\). Also, using an unpublished result of Jaco (reproven here by the authors) it is shown that if \(\pi_ 1(M)\) contains a genus \(g\) surface group, and \(H_ 1(M)\) has rank at least \(2g+2\), then \(M\) has a finite cover which is sufficiently large. When \(\pi_ 1(M)\) contains a free abelian subgroup of rank \(2\), the authors obtain various sharpenings of the torus theorem.
A second direction of applications is to growth rates of fundamental groups of 3-manifolds. For example, if \(M\) is closed hyperbolic, and the rank of the rational first homology is at least one, then for any \(K<3^{1/4}\), the number of words of length \(n\) (for any fixed generating set) is at least \(K^ n\) for all sufficiently large \(n\).
Another consequence of the underlying result is that if \(M\) is closed hyperbolic and \(H_ 1(M)\) has rank at least \(4\), any two noncommuting elements in \(\pi_ 1(M)\) generate a free subgroup. In a final section the authors use this to obtain lower bounds for the Margulis number, and hence the volumes, of hyperbolic 3-manifolds having enough homology.

MSC:

57M05 Fundamental group, presentations, free differential calculus
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
20F05 Generators, relations, and presentations of groups
20F14 Derived series, central series, and generalizations for groups
57M10 Covering spaces and low-dimensional topology
20E26 Residual properties and generalizations; residually finite groups
53C20 Global Riemannian geometry, including pinching
57M07 Topological methods in group theory
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