Isospectrality and commensurability of arithmetic hyperbolic
              2- and 3-manifolds

Reid, Alan W.

doi:10.1215/s0012-7094-92-06508-2

Cited by 69 publications

(82 citation statements)

References 16 publications

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“…In particular this gives affirmative answers to Questions 2.1, 2.2, 2.3 and 2.5 when M 1 and M 2 are both arithmetic. Indeed, Theorem 5.1 is a mild extension of the result in [19], and was discussed in [20]. …”

Section: On Questions 21 and 23mentioning

confidence: 98%

“…In this section we will focus on arithmetic hyperbolic manifolds in dimensions 2 and 3, discussing ideas in the proofs of the following results proved in [19] and [4] respectively. In particular this gives affirmative answers to Questions 2.1, 2.2, 2.3 and 2.5 when M 1 and M 2 are both arithmetic.…”

Section: On Questions 21 and 23mentioning

confidence: 99%

“…However, a slightly different version of this is required to finish the proof of Theorem 5.1. The key point to be proved (see [19] for details) is: Given this, the proof is completed as follows. Suppose L ∈ N 1 .…”

Section: 2mentioning

confidence: 99%

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Traces, lengths, axes and commensurability

Reid¹

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: On Questions 21 and 23mentioning

confidence: 98%

Section: On Questions 21 and 23mentioning

confidence: 99%

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Traces, lengths, axes and commensurability

Reid¹

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…A.W. Reid et C. Maclachlan [28], [29,1992] ont apporté une réponse affirmative à cette question dans le cas des surfaces fermées arithmétiques mais le cas général reste en suspend.…”

Section: Variantesunclassified

Le spectre des longueurs des surfaces hyperboliques : un exemple de rigidité.

Philippe¹

2010

Séminaire De Théorie Spectrale Et Géométrie

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Le spectre des longueurs des surfaces hyperboliques: un exemple de rigidité. Emmanuel PhilippeRésumé. -Après avoir présenté quelques résultats récents portant sur l'étude du spectre des longueurs des surfaces hyperboliques avec ou sans singularités, on démontre que les sphères possédant trois points coniques sont, dans leur classe, spectralement rigides.Abstract. -Having presented some recent results concerning the lentgh spectra of hyperbolic surfaces, we give a proof of the fact that spheres with three conical points are, in their class, spectrally rigid.Dans cet article, on fait agir un sous-groupe discret Γ de PSL 2 (R) par homographies sur le demi-plan de Poincaré H et on étudie les géodésiques fermées du quotient S = Γ\H. On renvoie à [3] pour toutes considérations élémentaires sur le plan hyperbolique et ses isométries.Le spectre des longueurs de Γ est l'ensemble des longueurs des élé-ments hyperboliques de Γ quand on parcourt l'ensemble des classes de conjugaison de tels éléments dans Γ. On ordonne ensuite cet ensemble dans l'ordre croissant en tenant compte des multiplicités.Si le quotient est compact le spectre est un ensemble discret dont le plus petit élément est appelé la systole. Deux problèmes se posent alors naturellement P1 : Déterminer, le groupe Γ étant donné, les premières valeurs du spectre. P2 : Le spectre Lsp Γ caractérise-t-il S à isométrie près ? Nous proposons tout d'abord dans ce papier un panorama des principaux résultats obtenus sur ces questions avant d'exposer un des rares exemples de rigidité spectrale connu.Concernant la première question, nous pouvons dores et déjà signaler que la détermination d'une formule simple pour la systole est un problème

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“…This normal space is isomorphic to R 2 , thus an orthogonal transformation is simply a rotation by an angle θ. Now the "complex length" of γ is defined to be s + iθ ( [7]). …”

Section: 1mentioning

confidence: 99%

On Elliptic Curves in SL₂(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

Winkelmann¹

2004

Nagoya Mathematical Journal

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Abstract. Let Γ be a discrete cocompact subgroup of SL2(C). We conjecture that the quotient manifold X = SL2(C)/Γ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where Γ∩SL2(R) is cocompact in SL2(R).Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2-and 3-folds. §1. Introduction Let Γ be a discrete cocompact subgroup of SL 2 (C). We are interested in closed complex analytic subspaces of the complex quotient manifold X = SL 2 (C)/Γ. It is well-known that X contains no hypersurfaces and it is easy to show that it contains no curves of genus 0. The existence of curves of genus ≥ 2 is an unsolved problem.On the other hand, it is not hard to show that there do exist curves of genus one (elliptic curves). (For these assertions, see [3], [9].) Our goal is to investigate how many different curves of genus one can be embedded in one such quotient manifold. There are only countably many abelian varieties which can be embedded into a quotient manifold of a complex semisimple Lie group by a discrete cocompact subgroup ([9, Cor. 4.6.2]). Thus the question is: Is the number of non-isomorphic elliptic curves in such a quotient SL 2 (C)/Γ finite or countably infinite?

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Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds

Cited by 69 publications

References 16 publications

Traces, lengths, axes and commensurability

Traces, lengths, axes and commensurability

Le spectre des longueurs des surfaces hyperboliques : un exemple de rigidité.

On Elliptic Curves in SL₂(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

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Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds

Cited by 69 publications

References 16 publications

Traces, lengths, axes and commensurability

Traces, lengths, axes and commensurability

Le spectre des longueurs des surfaces hyperboliques : un exemple de rigidité.

On Elliptic Curves in SL2(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

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Product

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On Elliptic Curves in SL₂(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths