Let be a convex co-compact discrete group of isometries of the hyperbolic plane , and the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian for large degree covers of given by where is a finite index normal subgroup of . Using techniques of thermodynamical formalism and representation theory, we prove two new existence results of sharp non-trivial resonances close to , in the large degree limit, for abelian covers and infinite index congruence subgroups of .
On considère ici des quotients du plan hyperbolique par des groupes d’isométries convexes co-compacts . On s’intéresse au comportement des résonances du Laplacien où est un revêtement Galoisien de haut degré de . 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 pour deux familles de revêtements, les cas abéliens et le cas des congruences.
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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
@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/} }
TY - JOUR AU - Jakobson, Dmitry AU - Naud, Frédéric AU - Soares, Louis TI - Large degree covers and sharp resonances of hyperbolic surfaces JO - Annales de l'Institut Fourier PY - 2020 SP - 523 EP - 596 VL - 70 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3319/ DO - 10.5802/aif.3319 LA - en ID - AIF_2020__70_2_523_0 ER -
%0 Journal Article %A Jakobson, Dmitry %A Naud, Frédéric %A Soares, Louis %T Large degree covers and sharp resonances of hyperbolic surfaces %J Annales de l'Institut Fourier %D 2020 %P 523-596 %V 70 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3319/ %R 10.5802/aif.3319 %G en %F AIF_2020__70_2_523_0
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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