## Duke Mathematical Journal

### Covering spaces of $3$-orbifolds

Marc Lackenby

#### 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 ${\rm PSL}(2,\mathbb{C})$ with torsion.) Then we prove that $O$ has a tower of finite-sheeted covers $\{O_i\}$ 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 $O_i$ 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 $\{O_i\}$ 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

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

#### 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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