Large degree covers and sharp resonances of hyperbolic surfaces
[Revêtements de haut degré et résonances des surfaces hyperboliques]
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 523-596.

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.

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 ().

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DOI : https://doi.org/10.5802/aif.3319
Classification : 58J50,  37C30,  37C35
Mots 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
@article{AIF_2020__70_2_523_0,
     author = {Jakobson, Dmitry and Naud, Fr\'ed\'eric and Soares, Louis},
     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},
     volume = {70},
     number = {2},
     year = {2020},
     doi = {10.5802/aif.3319},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3319/}
}
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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