Duke Mathematical Journal

Covering spaces of 3-orbifolds

Marc Lackenby

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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

Article information

Source
Duke Math. J., Volume 136, Number 1 (2007), 181-203.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1165244883

Digital Object Identifier
doi:10.1215/S0012-7094-07-13616-0

Mathematical Reviews number (MathSciNet)
MR2271299

Zentralblatt MATH identifier
1109.57015

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 30F40: Kleinian groups [See also 20H10] 20E07: Subgroup theorems; subgroup growth

Citation

Lackenby, Marc. Covering spaces of $3$ -orbifolds. Duke Math. J. 136 (2007), no. 1, 181--203. doi:10.1215/S0012-7094-07-13616-0. https://projecteuclid.org/euclid.dmj/1165244883


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