Families of mutually isospectral Riemannian orbifolds

Linowitz, Benjamin

doi:10.1112/blms/bdu096

Cited by 4 publications

(4 citation statements)

References 20 publications

(27 reference statements)

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“…Now, we have the trivial bound ω 2 (B) ≤ n k and the inequality h k ≤ 242d 3 4 k found in [45,Lemma 3.1]. Coupling these two inequalities with (19) produces…”

Section: 1mentioning

confidence: 59%

“…which lies in the length set (resp., complex length set) of exactly one of H 2 /Γ O , H 2 /Γ O 1 (resp., H 3 /Γ O , H 3 /Γ O 1 ). We note that the latter inequality (24) follows from the proof of [45,Thm 4.1]. By choosing constants appropriately, we contradict our hypothesis that the length set (resp., complex length sets) of H 2 /Γ O 1 and H 2 /Γ O 2 (or H 3 /Γ O 1 and H 3 /Γ O 2 ) coincide for all sufficiently small lengths.…”

Section: 2mentioning

confidence: 69%

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Counting and effective rigidity in algebra and geometry

et al. 2018

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The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

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“…Now, we have the trivial bound ω 2 (B) ≤ n k and the inequality h k ≤ 242d 3 4 k found in [45,Lemma 3.1]. Coupling these two inequalities with (19) produces…”

Section: 1mentioning

confidence: 59%

Section: 2mentioning

confidence: 69%

Counting and effective rigidity in algebra and geometry

et al. 2018

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“…It is worth noting that this lower bound grows faster than any polynomial function of x. In contrast to this, it was shown in [16] that the number of different isospectral Riemann surfaces which can be constructed by Vignéras' method is bounded above by a polynomial function of volume. In [17], McReynolds extended the lower bound of [7] to the groups of isometries of higher dimensional real hyperbolic spaces and complex hyperbolic 2-space.…”

Section: Introductionmentioning

confidence: 89%

Counting isospectral manifolds

Belolipetsky

Linowitz

2017

Advances in Mathematics

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ABSTRACT. Given a simple Lie group H of real rank at least 2 we show that the maximum cardinality of a set of isospectral non-isometric H-locally symmetric spaces of volume at most x grows at least as fast as x c log x/(log log x) 2 where c = c(H) is a positive constant. In contrast with the real rank 1 case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [3]. Our proof uses Sunada's method, results of [3], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.

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“…k . Indeed, the first bound follows since k A is contained in the narrow class field (whose degree is bounded above by the class number) and the second bound comes from work of Linowitz [22,Lemma 3.1]. As d k depends only on the commensurability class of Λ, we are reduced to showing that r f and |S| behave logarithmically with respect to volume.…”

Section: Arithmetic Lattice Boundsmentioning

confidence: 99%

Effective virtual and residual properties of some arithmetic hyperbolic 3–manifolds

DeBlois

Miller

Patel

2020

Trans. Amer. Math. Soc.

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We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth.

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Families of mutually isospectral Riemannian orbifolds

Cited by 4 publications

References 20 publications

Counting and effective rigidity in algebra and geometry

Counting and effective rigidity in algebra and geometry

Counting isospectral manifolds

Effective virtual and residual properties of some arithmetic hyperbolic 3–manifolds

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