Volume 10, issue 4 (2006)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
A random tunnel number one 3–manifold does not fiber over the circle

Nathan M Dunfield and Dylan P Thurston

Geometry & Topology 10 (2006) 2431–2499

arXiv: math.GT/0510129


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.

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
random 3-manifolds, tunnel number, interval exchanges, one-relator groups
Mathematical Subject Classification 2000
Primary: 57R22
Secondary: 57N10, 20F05
Forward citations
Received: 8 April 2006
Accepted: 13 November 2006
Published: 15 December 2006
Proposed: Cameron Gordon
Seconded: Rob Kirby, Joan Birman
Nathan M Dunfield
Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
Dylan P Thurston
Barnard College
Columbia University MC 4436
New York, NY 10027