A random tunnel number one 3-manifold does not fiber over the circle

Details A random tunnel number one 3-manifold does not fiber over the circle A random tunnel number one 3-manifold does not fiber over the circle

2005-10-06

arxiv.org/abs/math/0510129v4

Geom. Topol. 10 (2006) 2431-2499

Mathematics

Geometric Topology

This is the version published by Geometry & Topology on 15 December 2006

Scientific paper

10.2140/gt.2006.10.2431

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.

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