Large degree covers and sharp resonances of hyperbolic surfaces

Jakobson, Dmitry; Naud, Frédéric; Soares, Louis

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

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 ${\Delta }_{\tilde{X}}$ où $\tilde{X}=\tilde{\Gamma }\setminus {ℍ}^2$ 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 $\left\{\mathrm{Re}\right(s)=\delta \}$ pour deux familles de revêtements, les cas abéliens et le cas des congruences.

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 ${\Delta }_{\tilde{X}}$ for large degree covers of $X$ given by $\tilde{X}=\tilde{\Gamma }\setminus {ℍ}^2$ where $\tilde{\Gamma }⊲\Gamma$ 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 $\left\{\mathrm{Re}\right(s)=\delta \}$, in the large degree limit, for abelian covers and infinite index congruence subgroups of ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$.

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

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