Large covers and sharp resonances of hyperbolic surfaces

2017/10/16 by Dmitry Jakobson, Jakobson, Dmitry, Frederic Naud +3 · 1 citation
Mathematics · #FOS: Mathematics #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.1710.05666

This paper merges previous arXiv:1704.08546 and arXiv:1709.00295 in a unified setting with some improvements. Added more references and details

arxiv created 2017/11/17 · arxiv updated 2017/11/20

Abstract

Let Γ be a convex co-compact discrete group of isometries of the hyperbolic plane ℍ², and X=Γ\backslash ℍ² the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian for large degree covers of X given by a finite index normal subgroup of Γ. Using various techniques of thermodynamical formalism and representation theory, we prove two new existence results of "sharp non-trivial resonances" close to \Re(s)=δ_Γ, both in the large degree limit, for abelian covers and also infinite index congruence subgroups of SL2(ℤ).

Citations

• L-functions and sharp resonances of infinite index congruence subgroups of SL₂(ℤ)
• Hyperbolic Surfaces with Arbitrarily Small Spectral Gap

Cited by

• Spectral gaps and abelian covers of convex co-compact surfaces

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