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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
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.
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.
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