Computing arithmetic Kleinian groups

Page, Aurel

doi:10.1090/s0025-5718-2015-02939-1

Cited by 25 publications

(32 citation statements)

References 23 publications

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“…In this subsection we will see a shortcut to a method of computing Dirichlet domains introduced by Page [21], for Macfarlane manifolds. We focus on Page's algorithm because earlier ones have been specific to the non-compact arithmetic case [13,4,26,25], or have required either arithmeticity [8,29,21] or compactness [9,18] whereas, as we showed in §4, Macfarlane manifolds include examples from every combination of arithmetic and non-arithmetic with compact and non-compact.…”

Section: Quaternion Dirichlet Domainsmentioning

confidence: 99%

“…Thus, in our setup, Page's algorithm translates to searching for points in the orbit Orb Γ p1q " tγγ : | γ P Γu in order of increasing trace, and computing the perpendicular bisector of each one until all sides of the Dirichlet domain D Γ p1q have been found. Details and efficiency analysis of this can be found in [21].…”

Section: Quaternion Dirichlet Domainsmentioning

confidence: 99%

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Macfarlane hyperbolic 3–manifolds

Quinn

2018

Algebr. Geom. Topol.

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We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes of them arise in diverse topological and arithmetic settings. We then use this perspective to introduce a new method for computing their Dirichlet domains. We give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds. Proposition 2.2. [30, §3.3] If K Ă C, then D a faithful matrix representation of´a ,b Kī nto M 2 pCq. Moreover, under any such representation, n and tr correspond to the matrix determinant and trace, respectively.

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Section: Quaternion Dirichlet Domainsmentioning

confidence: 99%

Section: Quaternion Dirichlet Domainsmentioning

confidence: 99%

Macfarlane hyperbolic 3–manifolds

Quinn

2018

Algebr. Geom. Topol.

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“…In order to find many such G-coverings of every manifold Y = H 3 /Γ obtained in Section 5.1, we first compute a finite presentation of Γ, using the algorithms of [24]. Since Γ ∼ = π 1 (Y ), G-coverings of Y correspond to surjective homomorphisms Γ → G. Using the presentation of Γ, we enumerate surjective homomorphisms Γ → G up to conjugacy, using the methods described in [17, §9.1].…”

Section: 2mentioning

confidence: 99%

Torsion homology and regulators of isospectral manifolds

Bartel

Page

2016

J Topology

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Abstract. Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M1 and M2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. Here, we investigate the relationship between their integral homology. The Cheeger-Müller Theorem implies that a certain product of orders of torsion homology and of regulators for M1 agrees with that for M2. We exhibit a connection between the torsion in the integral homology of M1 and M2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg i (M1)/ Reg i (M2) representation theoretically. Further, we prove that the p ∞ -torsion in the homology of M1 is isomorphic to that of M2 for all primes p #G. For p ≤ 71, we give examples of pairs of strongly isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.

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“…There is further interest in homology with twisted coefficients, congruence subgroups and modular forms (see, for instance, [28,30]). Currently, Page [16] is working on optimizing algorithms in order to obtain more cell complexes for Bianchi groups and other Kleinian groups.…”

Section: The Algorithm Computing the Bianchi Fundamental Polyhedronmentioning

confidence: 99%

Higher torsion in the Abelianization of the full Bianchi groups

Rahm¹

2013

LMS J. Comput. Math.

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Denote by Q( √ −m), with m a square-free positive integer, an imaginary quadratic number field, and by O−m its ring of integers. The Bianchi groups are the groups SL2(O−m). In the literature, so far there have been no examples of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80 737 at the discriminant −1747.

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Computing arithmetic Kleinian groups

Abstract: Arithmetic Kleinian groups are arithmetic lattices in PSL 2 (C). We present an algorithm that, given such a group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism. 2

Cited by 25 publications

References 23 publications

Macfarlane hyperbolic 3–manifolds

Macfarlane hyperbolic 3–manifolds

Torsion homology and regulators of isospectral manifolds

Higher torsion in the Abelianization of the full Bianchi groups

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