## 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 Also, using an unpublished result of Jaco (reproven here by the authors) it is shown that if $$\pi_ 1(M)$$ contains a genus $$g$$ surface group, and $$H_ 1(M)$$ has rank at least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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