Covering spaces of arithmetic 3-orbifolds
| dc.creator | Lackenby, Marc | |
| dc.creator | Long, Darren D. | |
| dc.creator | Reid, Alan W. | |
| dc.date | 2006-01-27 | |
| dc.date | 2008-04-09 | |
| dc.date.accessioned | 2026-07-07T09:31:08Z | |
| dc.date.available | 2026-07-07T09:31:08Z | |
| dc.description | Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has infinite virtual first Betti number and G has super-exponential subgroup growth. Our first result, which forms the basis for the entire paper, is that G is commensurable with a lattice containing the Klein group of order 4. Our second result is that if G has a finite index subgroup with first Betti number at least 4, then G is large. This is known to hold for many arithmetic lattices. In particular, we show that it is always the case when G contains A_4, S_4 or A_5. Our third main result is that the Lubotzky-Sarnak conjecture and the geometrisation conjecture together imply that any arithmetic Kleinian group G is large. We also give a new 'elementary' proof that arithmetic Kleinian groups do not have the congruence subgroup property, which avoids the use of the Golod-Shafarevich inequality. | |
| dc.description | 41 pages; v2: minor changes; to appear in IMRN | |
| dc.identifier | https://arxiv.org/abs/math/0601677 | |
| dc.identifier | http://arxiv.org/abs/math/0601677 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158356 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57N10 (Primary); 57M10, 57M50, 11F75 (Secondary) | |
| dc.title | Covering spaces of arithmetic 3-orbifolds | |
| dc.type | text |