## Geometry & Topology

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

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 10, Number 4 (2006), 2431-2499.

Dates
Accepted: 13 November 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799807

Digital Object Identifier
doi:10.2140/gt.2006.10.2431

Mathematical Reviews number (MathSciNet)
MR2284062

Zentralblatt MATH identifier
1139.57018

#### Citation

Dunfield, Nathan M; Thurston, Dylan P. A random tunnel number one 3–manifold does not fiber over the circle. Geom. Topol. 10 (2006), no. 4, 2431--2499. doi:10.2140/gt.2006.10.2431. https://projecteuclid.org/euclid.gt/1513799807

#### References

• I Agol, J Hass, W Thurston, The computational complexity of knot genus and spanning area, Trans. Amer. Math. Soc. 358 (2006) 3821–3850
• J Berge, Documentation for the program Heegaard, preprint (c1990) Available at \setbox0\makeatletter\@url http://www.math.uic.edu/~t3m/ {\unhbox0
• R Bieri, W D Neumann, R Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987) 451–477
• J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)
• A Borisov, M Sapir, Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms
• K S Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987) 479–504
• J O Button, Fibred and virtually fibred hyperbolic 3-manifolds in the censuses, Experiment. Math. 14 (2005) 231–255
• P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$-manifolds, with microfiche supplement, Math. Comp. 68 (1999) 321–332
• N M Dunfield, Alexander and Thurston norms of fibered 3-manifolds, Pacific J. Math. 200 (2001) 43–58
• N M Dunfield, W P Thurston, Finite covers of random 3-manifolds, Invent. Math 166 (2006) 457–521
• M Dwass, Simple random walk and rank order statistics, Ann. Math. Statist. 38 (1967) 1042–1053
• M Gromov, Hyperbolic groups, from: “Essays in group theory”, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75–263
• J Hoste, M Thistlethwaite, Knotscape (1999) Available at \setbox0\makeatletter\@url http://www.math.utk.edu/~morwen {\unhbox0
• J Hoste, M Thistlethwaite, J Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998) 33–48
• W Jaco, }, Proc. Amer. Math. Soc. 92 (1984) 288–292
• W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed $3$-manifold, Illinois J. Math. 39 (1995) 358–406
• V A Kaimanovich, Poisson boundary of discrete groups, preprint (2001)
• V A Kaimanovich, H Masur, The Poisson boundary of Teichmüller space, J. Funct. Anal. 156 (1998) 301–332
• S P Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergodic Theory Dynam. Systems 5 (1985) 257–271
• C J Leininger, Surgeries on one component of the Whitehead link are virtually fibered, Topology 41 (2002) 307–320
• F Luo, R Stong, Dehn-Thurston coordinates for curves on surfaces, Comm. Anal. Geom. 12 (2004) 1–41
• H Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems 6 (1986) 99–116
• M Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, preprint (2004) to appear in Ann. of Math.
• K Murasugi, }, Amer. J. Math. 85 (1963) 544–550
• A Y Ol'shanskiĭ, Almost every group is hyperbolic, Internat. J. Algebra Comput. 2 (1992) 1–17
• R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992)
• D Poulalhon, G Schaeffer, Optimal coding and sampling of triangulations, from: “Automata, languages and programming”, Lecture Notes in Comput. Sci. 2719, Springer, Berlin (2003) 1080–1094
• S Rankin, O Flint, J Schermann, Enumerating the prime alternating knots. I, J. Knot Theory Ramifications 13 (2004) 57–100
• S Rankin, O Flint, J Schermann, Enumerating the prime alternating knots. II, J. Knot Theory Ramifications 13 (2004) 101–149
• G Schaeffer, P Zinn-Justin, On the asymptotic number of plane curves and alternating knots, Experiment. Math. 13 (2004) 483–493
• M Scharlemann, Outermost forks and a theorem of Jaco, from: “Combinatorial methods in topology and algebraic geometry (Rochester, N.Y. 1982)”, Contemp. Math. 44, Amer. Math. Soc., Providence, RI (1985) 189–193
• S Schleimer, Almost normal Heegaard splittings, PhD thesis, Berkeley (2001)
• J Stallings, On fibering certain $3$-manifolds, from: “Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)”, Prentice-Hall (1962) 95–100
• J M Steele, The Bohnenblust-Spitzer algorithm and its applications, Probabilistic methods in combinatorics and combinatorial optimization, J. Comput. Appl. Math. 142 (2002) 235–249
• J L Tollefson, N Wang, Taut normal surfaces, Topology 35 (1996) 55–75
• G S Walsh, Great circle links and virtually fibered knots, Topology 44 (2005) 947–958
• J Weeks, SnapPea Available at \setbox0\makeatletter\@url http://www.geometrygames.org {\unhbox0