Huber's Theorem for Hyperbolic Orbisurfaces

Dryden, Emily B.; Strohmaier, Alexander

doi:10.4153/cmb-2009-008-0

Cited by 14 publications

(14 citation statements)

References 11 publications

(13 reference statements)

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“…It is not known whether the statement concerning nonisospectrality of manifolds and orbifolds remains true without the condition of a common Riemannian covering. Dryden and Strohmaier [9] showed that on oriented compact hyperbolic orbifolds in dimension two, the spectrum completely determines the types and numbers of singular points. Independently, this had also been shown by the first author together with P.G.…”

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confidence: 99%

Isospectral orbifolds with different maximal isotropy orders

2008

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We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on p-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean R 3 and are shown to be isospectral on functions using dimension formulas for the eigenspaces developed in [12]. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties.

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confidence: 99%

Isospectral orbifolds with different maximal isotropy orders

2008

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“…By Theorem 2.2, since X(Γ) and X(Γ ) are representation equivalent, they are isospectral with respect to the Laplace operator. When dim(X(Γ)) = dim(X(Γ )) = 2, it follows that X(Γ) and X(Γ ) are length isospectral by work of Dryden and Strohmaier [34] (see also Doyle and Rossetti [32,33]), so in particular L(X(Γ)) = L(X(Γ )). When dim(X(Γ)) = dim(X(Γ )) = 3, it follows from the trace formula of Duistermaat and Guillemin [36] (see also Prasad and Rapinchuk [72, Theorem 10.1], and in the orbifold case Elstrodt, Grunewald, and Mennicke [39, p. 203]) that X(Γ) and X(Γ ) have the same sets of lengths of closed geodesics, considered without multiplicity.…”

Section: Isometriesmentioning

confidence: 99%

“…However, this procedure changes the spectrum, and indeed "one can hear the orders of the cone points of orientable hyperbolic 2-orbifold" [33]. For further discussion on this point, see also work of Dryden [34], Dryden, Gordon, Greenwald, and Webb [35,Section 5], and Gordon and Rossetti [49,Proposition 3.4]. )…”

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confidence: 99%

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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“…En ce qui concerne le rapport entre le spectre du Laplacien et le spectre des longueurs on renvoie aux travaux de W. Müller [21,1992] pour le cas non compact et à ceux de E. Dryden et A. Strohmaïer [14,2004], [15] pour le cas des orbifolds.…”

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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Huber's Theorem for Hyperbolic Orbisurfaces

Cited by 14 publications

References 11 publications

Isospectral orbifolds with different maximal isotropy orders

Isospectral orbifolds with different maximal isotropy orders

Small isospectral and nonisometric orbifolds of dimension 2 and 3

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

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