A census of cusped hyperbolic 3-manifolds

Callahan, Patrick J.; Hildebrand, Martin; Weeks, Jeffrey R.

doi:10.1090/s0025-5718-99-01036-4

Cited by 94 publications

(188 citation statements)

References 8 publications

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“…For instance, about 90% of the cusped manifolds in the census of Callahan, Hildebrand and Weeks [8] are fibered (Button [7]), and most of these manifolds have tunnel number one. Without the naive version of Brown's criterion, one would not be able to examine enough manifolds to suggest Theorem 2.4; without our improved train track version, we would not have come to the correct version of Conjecture 1.2.…”

Section: The Tyranny Of Small Examplesmentioning

confidence: 99%

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

Dunfield

Thurston

2006

Geom. Topol.

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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 between 0 and 1. 57R22; 57N10, 20F05We dedicate this paper to the memory of Raoul Bott (1923Bott ( -2005, a wise teacher and warm friend, always searching for the simplicity at the heart of mathematics.

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Section: The Tyranny Of Small Examplesmentioning

confidence: 99%

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

Dunfield

Thurston

2006

Geom. Topol.

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“…These manifolds are relevant to the theory because two of our bricks are homeomorphic to the complement of 6 3 1 (itself denoted by M 6 3 1 in [5]), with appropriate markings.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Geometric Three-manifolds

Martelli

Petronio

2004

Geometriae Dedicata

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We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M 6 3 1 (the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into 'bricks' of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.

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“…See Section 6.3 for an example of an ideal triangulation T of the census manifold m136 [3] which admits a semi-strict angle structure (i.e., angles are nonnegative real numbers), does not admit a strict angle structure, and which has a solution of the gluing equations that recover the complete hyperbolic structure. A case-by-case analysis shows that this example admits an index structure, thus the index I T exists.…”

Section: Definition 22 (A)mentioning

confidence: 99%

The 3D index of an ideal triangulation and angle structures

Garoufalidis

2016

Ramanujan J

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Abstract. The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with r torii boundary components. For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients.Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves.

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A census of cusped hyperbolic 3-manifolds

Cited by 94 publications

References 8 publications

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

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

Complexity of Geometric Three-manifolds

The 3D index of an ideal triangulation and angle structures

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