Large degree covers and sharp resonances of hyperbolic surfaces

Jakobson, Dmitry; Naud, Frédéric; Soares, Louis

doi:10.5802/aif.3319

Large degree covers and sharp resonances of hyperbolic surfaces
[Revêtements de haut degré et résonances des surfaces hyperboliques]

Jakobson, Dmitry ¹ ; Naud, Frédéric ² ; Soares, Louis ³

¹ McGill University Department of Mathematics and Statistics 805 Sherbrooke Street West Montreal, Quebec, H3A0B9 (Canada)
² Laboratoire de Mathématiques d’Avignon Avignon Université, Campus Jean-Henri Fabre, 301 rue Baruch de Spinoza 84916 Avignon Cedex 9 (France)
³ Friedrich-Schiller-Universität Jena Institut für Mathematik Ernst-Abbe-Platz 2, 07743 Jena (Germany)

Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 523-596.

Résumé
Abstract

On considère ici des quotients $X = Γ ∖ ℍ^{2}$ du plan hyperbolique $ℍ^{2}$ par des groupes d’isométries convexes co-compacts $Γ$ . On s’intéresse au comportement des résonances du Laplacien $Δ_{\tilde{X}}$ où $\tilde{X} = \tilde{Γ} ∖ ℍ^{2}$ est un revêtement Galoisien de haut degré de $X$ . En combinant des techniques de formalisme thermodynamique et de théorie des représentations, on prouve, dans le régime de haut degré, de nouveaux théorèmes d’existence de résonances non-triviales près de l’axe ${Re (s) = δ}$ pour deux familles de revêtements, les cas abéliens et le cas des congruences.

Let $Γ$ be a convex co-compact discrete group of isometries of the hyperbolic plane $ℍ^{2}$ , and $X = Γ ∖ ℍ^{2}$ the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian $Δ_{\tilde{X}}$ for large degree covers of $X$ given by $\tilde{X} = \tilde{Γ} ∖ ℍ^{2}$ where $\tilde{Γ} ⊲ Γ$ is a finite index normal subgroup of $Γ$ . Using techniques of thermodynamical formalism and representation theory, we prove two new existence results of sharp non-trivial resonances close to ${Re (s) = δ}$ , in the large degree limit, for abelian covers and infinite index congruence subgroups of ${SL}_{2} (ℤ)$ .

Reçu le : 2018-07-03
Révisé le : 2019-02-04
Accepté le : 2019-03-12
Publié le : 2020-05-28

DOI : 10.5802/aif.3319

Classification : 58J50, 37C30, 37C35
Keywords: Hyperbolic surfaces, Geometrically finite fuchsian groups, Laplace spectrum and resonances, Selberg zeta function, Representation theory, Transfer operators and thermodynamical formalism
Mot clés : Surfaces hyperboliques, groupes Fuchsiens géométriquement finis, Spectre du Laplacien et résonances, fonctions zêtas de Selberg, théorie des représentations, opérateurs de transfert et formalisme thermodynamique

Affiliations des auteurs :

Jakobson, Dmitry ¹ ; Naud, Frédéric ² ; Soares, Louis ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Large degree covers and sharp resonances of hyperbolic surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {523--596},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Jakobson, Dmitry; Naud, Frédéric; Soares, Louis. Large degree covers and sharp resonances of hyperbolic surfaces. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 523-596. doi : 10.5802/aif.3319. https://aif.centre-mersenne.org/articles/10.5802/aif.3319/

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