Covering spaces of 3-orbifolds
| dc.creator | Lackenby, Marc | |
| dc.date | 2005-08-01 | |
| dc.date | 2006-06-07 | |
| dc.date.accessioned | 2026-07-07T06:42:43Z | |
| dc.date.available | 2026-07-07T06:42:43Z | |
| dc.description | Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, 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 {O_i} with linear growth of 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. Firstly, the fundamental group of O has at least exponential subgroup growth. Secondly, the covers {O_i} have positive Heegaard gradient. Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has 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 non-empty singular locus has large fundamental group. Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds. | |
| dc.description | 26 pages. Version 3 has only minor changes from versions 1 and 2. To appear in Duke Mathematical Journal | |
| dc.identifier | https://arxiv.org/abs/math/0508033 | |
| dc.identifier | http://arxiv.org/abs/math/0508033 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102145 | |
| dc.subject | Geometric Topology | |
| dc.subject | Group Theory | |
| dc.subject | 57N10; 30F40; 20E07 | |
| dc.title | Covering spaces of 3-orbifolds | |
| dc.type | text |