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A random tunnel number one 3manifold does not fiber over the circle
 Geom. Topol
"... 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 exp ..."
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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
INCREASING THE NUMBER OF FIBERED FACES OF ARITHMETIC HYPERBOLIC 3MANIFOLDS
"... Abstract. We exhibit a closed hyperbolic 3manifold which satisfies a very strong form of Thurston’s Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of ..."
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Abstract. We exhibit a closed hyperbolic 3manifold which satisfies a very strong form of Thurston’s Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed numbertheoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried’s dynamical characterization of the fibered faces. The origin of the basic fibration M → S1 is the modular elliptic curve E = X0(49), which admits multiplication by the ring of integers of Q [ √ −7]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K = Q [ √ −3], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of O ∗ D of level 7. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finitevolume hyperbolic 3manifold with the same properties by using a
The Thurston norm via Normal Surfaces
, 2007
"... To Bill Thurston on the occasion of his sixtieth birthday Abstract Given a triangulation of a closed, oriented, irreducible, atoroidal 3–manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded ..."
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To Bill Thurston on the occasion of his sixtieth birthday Abstract Given a triangulation of a closed, oriented, irreducible, atoroidal 3–manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by nonnegative integer weights, 14 for each 3–simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, B, of the Thurston norm. Applications of this approach include (1) an algorithm to compute B and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of B in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of B and hence an algorithm to decide whether a 3–manifold fibres over the circle. AMS Classification 57M25, 57N10
Normalizing HeegaardScharlemannThompson Splittings
"... Abstract. We define a HeegaardScharlemannThompson (HST) splitting of a 3manifold M to be a sequence of pairwisedisjoint, embedded surfaces, {Fi}, such that for each odd value of i, Fi is a Heegaard splitting of the submanifold of M cobounded by Fi−1 and Fi+1. Our main result is the following: Su ..."
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Abstract. We define a HeegaardScharlemannThompson (HST) splitting of a 3manifold M to be a sequence of pairwisedisjoint, embedded surfaces, {Fi}, such that for each odd value of i, Fi is a Heegaard splitting of the submanifold of M cobounded by Fi−1 and Fi+1. Our main result is the following: Suppose M (̸ = B 3 or S 3) is an irreducible submanifold of a triangulated 3manifold, bounded by a normal or almost normal surface, and containing at most one maximal normal 2sphere. If {Fi} is a strongly irreducible HST splitting of M then we may isotope it so that for each even value of i the surface Fi is normal and for each odd value of i the surface Fi is almost normal. We then show how various theorems of Rubinstein, Thompson, Stocking and Schleimer follow from this result. We also show how our results imply the following: (1) a manifold that contains a nonseparating surface contains an almost normal one, and (2) if a manifold contains a normal Heegaard surface then it contains two almost normal ones that are topologically parallel to it.
TIGHTENING ALMOST NORMAL SURFACES
"... Abstract. We present a specialized version of Haken’s normalization procedure. Our main theorem states that there is a compression body canonically associated to a given transversely oriented almost normal surface. Several applications are given. 1. ..."
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Abstract. We present a specialized version of Haken’s normalization procedure. Our main theorem states that there is a compression body canonically associated to a given transversely oriented almost normal surface. Several applications are given. 1.
Strongly Irreducible Surface Automorphisms
"... A surface automorphism is strongly irreducible if every essential simple closed curve in the surface intersects its image nontrivially. We show that a threemanifold admits only finitely many surface bundle structures with strongly irreducible monodromy. 1. ..."
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A surface automorphism is strongly irreducible if every essential simple closed curve in the surface intersects its image nontrivially. We show that a threemanifold admits only finitely many surface bundle structures with strongly irreducible monodromy. 1.
SPHERE RECOGNITION LIES IN NP
"... Abstract. We prove that the threesphere recognition problem lies in the complexity class NP. Our work relies on Thompson’s original proof that the problem is decidable [Math. Res. Let., 1994], Casson’s version of her algorithm, and recent results of Agol, Hass, and Thurston [STOC, 2002]. 1. ..."
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Abstract. We prove that the threesphere recognition problem lies in the complexity class NP. Our work relies on Thompson’s original proof that the problem is decidable [Math. Res. Let., 1994], Casson’s version of her algorithm, and recent results of Agol, Hass, and Thurston [STOC, 2002]. 1.
THE DISJOINT ANNULUS PROPERTY
"... Abstract. A Heegaard splitting of a closed, orientable threemanifold satisfies the Disjoint Annulus Property if each handlebody contains an essential annulus and these are disjoint. This paper proves that, for a fixed threemanifold, all but finitely many splittings have the disjoint annulus propert ..."
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Abstract. A Heegaard splitting of a closed, orientable threemanifold satisfies the Disjoint Annulus Property if each handlebody contains an essential annulus and these are disjoint. This paper proves that, for a fixed threemanifold, all but finitely many splittings have the disjoint annulus property. As a corollary, all but finitely many splittings have distance three or less, as defined by Hempel. 1. History and overview Great effort has been spent on the classification problem for Heegaard splittings of threemanifolds. Haken’s lemma [2], that all splittings of a reducible manifold are themselves reducible, could be considered one of the first results in this direction. Weak reducibility was introduced by Casson and Gordon [1] as a generalization of reducibility. They concluded that a weakly reducible splitting is either itself reducible or the manifold in question contains an incompressible surface. Thompson [16] later defined the disjoint curve property as a further generalization of weak reducibility. She deduced that all splittings of a toroidal threemanifold have the