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15 January 2007 Covering spaces of 3-orbifolds
Marc Lackenby
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Duke Math. J. 136(1): 181-203 (15 January 2007). DOI: 10.1215/S0012-7094-07-13616-0

Abstract

Let O be a compact orientable 3-orbifold with nonempty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {Oi} with linear growth of mod p homology for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of Oi grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. First, the fundamental group of O has at least exponential subgroup growth. Second, the covers {Oi} have positive Heegaard gradient. Third, we use the existence of this tower of covers to show that a group-theoretic conjecture of Lubotzky and Zelmanov implies that O has a large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with nonempty singular locus has large fundamental group. Many of these results also apply to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with nonempty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank-two fundamental group and all arithmetic 3-manifolds

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Marc Lackenby. "Covering spaces of 3-orbifolds." Duke Math. J. 136 (1) 181 - 203, 15 January 2007. https://doi.org/10.1215/S0012-7094-07-13616-0

Information

Published: 15 January 2007
First available in Project Euclid: 4 December 2006

zbMATH: 1109.57015
MathSciNet: MR2271299
Digital Object Identifier: 10.1215/S0012-7094-07-13616-0

Subjects:
Primary: 20E07, 30F40, 57N10

Rights: Copyright © 2007 Duke University Press

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Vol.136 • No. 1 • 15 January 2007
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