Large degree covers and sharp resonances of hyperbolic surfaces
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 523-596.

Let Γ be a convex co-compact discrete group of isometries of the hyperbolic plane 2 , and X=Γ 2 the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian Δ X ˜ for large degree covers of X given by X ˜=Γ ˜ 2 where 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 ().

On considère ici des quotients X=Γ 2 du plan hyperbolique 2 par des groupes d’isométries convexes co-compacts Γ. On s’intéresse au comportement des résonances du Laplacien Δ X ˜ X ˜=Γ ˜ 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.

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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

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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jakobson, Dmitry; Naud, Frédéric; Soares, Louis. Large degree covers and sharp resonances of hyperbolic surfaces. Annales de l'Institut Fourier, Volume 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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