2006
DOI: 10.2140/gt.2006.10.2431
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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 numbe… Show more

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Cited by 32 publications
(26 citation statements)
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“…(1) We show in Lemma 6.1 that a group G satisfies the hypothesis of the theorem if there exists [φ] ∈ S(G) such that both [φ] and [−φ] lie in Σ(G). This is not as rare an occurrence as it might sound: Dunfield and D. Thurston [17,Section 6] give strong evidence for the conjecture that "most" groups with a nice (2, 1)-presentation have this property. (2) In [23] the authors and Kevin Schreve showed that if G is the fundamental group of an aspherical 3-manifold, then G is in fact residually G. The key ingredient in the proof is the fact that such groups are "virtually special" in the sense of Haglund-Wise [26].…”
Section: Stefan F S Tmentioning
confidence: 99%
“…(1) We show in Lemma 6.1 that a group G satisfies the hypothesis of the theorem if there exists [φ] ∈ S(G) such that both [φ] and [−φ] lie in Σ(G). This is not as rare an occurrence as it might sound: Dunfield and D. Thurston [17,Section 6] give strong evidence for the conjecture that "most" groups with a nice (2, 1)-presentation have this property. (2) In [23] the authors and Kevin Schreve showed that if G is the fundamental group of an aspherical 3-manifold, then G is in fact residually G. The key ingredient in the proof is the fact that such groups are "virtually special" in the sense of Haglund-Wise [26].…”
Section: Stefan F S Tmentioning
confidence: 99%
“…We can comb (See Section 1.4 of [25]) a non-classical interval exchange moving left to right along the base interval to get a generic train track. For example, see the first picture in Figure 21 of [5]. It can be directly checked that the resulting track is transversely recurrent.…”
Section: Sub-surface Projections To Carried Curvesmentioning
confidence: 99%
“…For reasons of homology, a random 3-manifold in the sense of Section 2 fibers over the circle with probability 0 (see Dunfield and Thurston [7,Corollary 8.5]) and even among those manifolds with b 1 > 0, surface bundles appear to be rare (see Dunfield and Thurston [6]). Despite this, we show that for a natural model of random bundles the distribution of jZ.M /j is the same as that of random manifolds more generally.…”
Section: Random Surface Bundlesmentioning
confidence: 99%