Duke Mathematical Journal
- Duke Math. J.
- Volume 136, Number 1 (2007), 181-203.
Covering spaces of -orbifolds
Let be a compact orientable -orbifold with nonempty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of is the quotient of hyperbolic -space by a lattice in with torsion.) Then we prove that has a tower of finite-sheeted covers with linear growth of mod homology for some prime . This means that the dimension of the first homology, with mod coefficients, of the fundamental group of grows linearly in the covering degree. The proof combines techniques from -manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about -adic analytic pro- groups. This has several consequences. First, the fundamental group of has at least exponential subgroup growth. Second, the covers 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 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 -orbifold with nonempty singular locus has large fundamental group. Many of these results also apply to -manifolds commensurable with an orientable finite-volume hyperbolic -orbifold with nonempty singular locus. This includes all closed orientable hyperbolic -manifolds with rank-two fundamental group and all arithmetic -manifolds
Duke Math. J., Volume 136, Number 1 (2007), 181-203.
First available in Project Euclid: 4 December 2006
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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