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Published December 15, 2006 | public
Journal Article Open

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


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.

Additional Information

© Copyright 2006 Geometry & Topology. Received: 8 April 2006. Accepted: 13 November 2006. Published: 15 December 2006. Proposed: Cameron Gordon; Seconded: Rob Kirby, Joan Birman. Dunfield was partially supported by the US National Science Foundation, both by grant #DMS-0405491 and as a Postdoctoral Fellow. He was also supported by a Sloan Fellowship, and some of the work was done while he was at Harvard University. Thurston was partially supported by the US National Science Foundation as a Postdoctoral Fellow. Most of the work was done while he was at Harvard University. The authors also thank Steve Kerckhoff for helpful conversations and correspondence, as well as the referee for their very careful reading of this paper and resulting detailed comments. We dedicate this paper to the memory of Raoul Bott (1923–2005), a wise teacher and warm friend, always searching for the simplicity at the heart of mathematics. Additional material for downloading: The first two items check certain combinatorial facts asserted in Section 10. The other two are the binary and source code for the genus2fiber program.


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