On Fields of Definition of Arithmetic Kleinian Reflection Groups II

Belolipetsky, Mikhail; Linowitz, Benjamin

doi:10.1093/imrn/rns292

Cited by 5 publications

(5 citation statements)

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“…It was shown there, in particular, that the degree of the field of definition of arithmetic reflection groups in dimension 3 is bounded above by 35. In a joint work with Linowitz [BL14], we improved this bound to 9, which essentially allows to give a list of all possible fields of definition (see [BL14] for the details). The case of n = 2 was considered by Maclachlan in [Mac11].…”

Section: Effective Results Obtained By the Spectral Methodsmentioning

confidence: 99%

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Arithmetic hyperbolic reflection groups

Belolipetsky

2016

Bull. Amer. Math. Soc.

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Section: Effective Results Obtained By the Spectral Methodsmentioning

confidence: 99%

“…The result of [Nik11], together with [Mac11] and [BL14] which we shall discuss later on in Section 5, implies:…”

Section: The Work Of Nikulinmentioning

confidence: 97%

Arithmetic hyperbolic reflection groups

Belolipetsky

2016

Bull. Amer. Math. Soc.

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“…The main hurdle is the invariant OEk A W k or rather OEk 0 A W k associated with A, for which we apparently miss good bounds and which we are forced to estimate by the class number. Some observations on this invariant have been also made by Belolipetsky and Linowitz [3] in connection with the enumeration of fields of definition of arithmetic Kleinian reflection groups.…”

Section: Remark 23 (1)mentioning

confidence: 87%

Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹

Džambić

2015

Geom. Topol.

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We study quotients nH n of the n-fold product of the upper half-plane H by irreducible and torsion-free lattices < PSL 2 .R/ n with the same Betti numbers as the n-fold product .P 1 / n of projective lines. Such varieties are called fake products of projective lines or fake .P 1 / n. These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake .P 1 / n is finite (independently of n), we give examples of fake .P 1 / 4 and show that for n > 4 there are no fake .P 1 / n of the form nH n with contained in the norm-one group of a maximal order of a quaternion algebra over a real number field. 11F06, 22E40

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“…Gabai, Meyerhoff, and Milley identified the Weeks manifold as the lowest volume 3-manifold [41,42], following work of Chinburg, Friedman, Jones, and Reid [21] who showed it was the smallest arithmetic 3-manifold. It arises from a subgroup of index 12 in a maximal group associated to a maximal order in the quaternion algebra over the cubic field of discriminant −23 (generated by a root of x 3 − x + 1) ramified at the real place and the prime of norm 5, or equivalently is obtained by (5, 1), (5,2) surgery on the two components of the Whitehead link. The Weeks manifold has volume 3 · 23 3/2 ζ F (2) 4π 4 = 0.9427 .…”

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confidence: 99%

“…These methods allow us to list all arithmetic groups of bounded volume, in particular reproving the results of Chinburg and Friedman and Chinburg, Friedman, Jones, and Reid cited above, replacing technical lemmas with a sequence of algorithmically checkable bounds. We expect that this blend of ideas will be useful in other domains: they have already been adapted to the case of arithmetic reflection groups by Belolipetsky and Linowitz [5]. Finally, we use algorithms for quaternion algebras and arithmetic Fuchsian and Kleinian groups, implemented in MAGMA [8], to carry out the remaining significant enumeration problem: we list the possible pairs and identify the ones of smallest area under congruence hypotheses for both orbifolds and manifolds in dimensions 2 and 3.…”

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Small isospectral and nonisometric orbifolds of dimension 2 and 3

Linowitz

Voight

2015

Math. Z.

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ABSTRACT. Revisiting a construction due to Vignéras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.Introduction. In 1966, Kac [54] famously posed the question: "Can one hear the shape of a drum?" In other words, if you know the frequencies at which a drum vibrates, can you determine its shape? Since this question was asked, hundreds of articles have been written on this general topic, and it remains a subject of considerable interest [45].Let (M, g) be a connected, compact Riemannian manifold (with or without boundary). Associated to M is the Laplace operator ∆, defined by) form an infinite, discrete sequence of nonnegative real numbers 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . , called the spectrum of M . In the case that M is a planar domain, the eigenvalues in the spectrum of M are essentially the frequencies produced by a drum shaped like M and fixed at its boundary. Two Riemannian manifolds are said to be Laplace isospectral if they have the same spectra. Inverse spectral geometry asks the extent to which the geometry and topology of M are determined by its spectrum. For example, volume, dimension and scalar curvature can all be shown to be spectral invariants.The problem of whether the isometry class itself is a spectral invariant received a considerable amount of attention beginning in the early 1960s. In 1964, Milnor [65] gave the first negative example, exhibiting a pair of Laplace isospectral and nonisometric 16-dimensional flat tori. In 1980, Vignéras [83] constructed non-positively curved, Laplace isospectral and nonisometric manifolds of dimension n for every n ≥ 2 (hyperbolic for n = 2, 3). The manifolds considered by Vignéras are locally symmetric spaces arising from arithmetic: they are quotients by discrete groups of isometries obtained from unit groups of orders in quaternion algebras over number fields. A few years later, Sunada [80] developed a general algebraic method for constructing Laplace isospectral manifolds arising from almost conjugate subgroups of a finite group of isometries acting on a manifold, inspired by the existence of number fields with the same zeta function where the group and its subgroups form what is known as a Gassmann triple [9]. Sunada's method is extremely versatile and, along with its variants, accounts for the majority of known constructions of Laplace isospectral non-isometric manifolds [30]. Indeed, in 1992, Gordon, Webb and Wolpert [50] used an extension of Sunada's method in order to answer Kac's question in the negative for planar surfaces: "One cannot hear the shape of a drum." For expositions of Sunada's method, the work it has influenced, and the many other constructions of Laplace isospectral manifolds, we refer the reader to surveys of Gordon [46,47] and the references therein.The examples of Vignéras are distinguished as they cannot arise from Sunada's method: in particular, they do not cover a common orbi...

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On Fields of Definition of Arithmetic Kleinian Reflection Groups II

Cited by 5 publications

References 22 publications

Arithmetic hyperbolic reflection groups

Arithmetic hyperbolic reflection groups

Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹

Small isospectral and nonisometric orbifolds of dimension 2 and 3

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On Fields of Definition of Arithmetic Kleinian Reflection Groups II

Cited by 5 publications

References 22 publications

Arithmetic hyperbolic reflection groups

Arithmetic hyperbolic reflection groups

Varieties of general type with the same Betti numbers as ℙ1× ℙ1×⋯× ℙ1

Small isospectral and nonisometric orbifolds of dimension 2 and 3

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Varieties of general type with the same Betti numbers as ℙ¹× ℙ¹×⋯× ℙ¹