A random tunnel number one 3-manifold does not fiber over the circle
| dc.creator | Dunfield, Nathan M | |
| dc.creator | Thurston, Dylan P | |
| dc.date | 2005-10-06 | |
| dc.date | 2009-03-03 | |
| dc.date.accessioned | 2026-07-07T12:48:15Z | |
| dc.date.available | 2026-07-07T12:48:15Z | |
| dc.description | We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1. | |
| dc.description | This is the version published by Geometry & Topology on 15 December 2006 | |
| dc.identifier | https://arxiv.org/abs/math/0510129 | |
| dc.identifier | http://arxiv.org/abs/math/0510129 | |
| dc.identifier | Geom. Topol. 10 (2006) 2431-2499 | |
| dc.identifier | doi:10.2140/gt.2006.10.2431 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/221994 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57R22, 20F05, 57N10 | |
| dc.title | A random tunnel number one 3-manifold does not fiber over the circle | |
| dc.type | text |