# ANNALES DE L'INSTITUT FOURIER

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

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

On considère ici des quotients $X=\Gamma \setminus {ℍ}^{2}$ du plan hyperbolique ${ℍ}^{2}$ par des groupes d’isométries convexes co-compacts $\Gamma$. On s’intéresse au comportement des résonances du Laplacien ${\Delta }_{\stackrel{˜}{X}}$$\stackrel{˜}{X}=\stackrel{˜}{\Gamma }\setminus {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 $\left\{\mathrm{Re}\left(s\right)=\delta \right\}$ pour deux familles de revêtements, les cas abéliens et le cas des congruences.

Revised:
Accepted:
Published online:
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
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)
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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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