Large degree covers and sharp resonances of hyperbolic surfaces

Abstract

Let Γ be a convex co-compact discrete group of isometries of the hyperbolic plane ℍ 2 , and X=Γ∖ℍ 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 ˜∖ℍ 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 (ℤ).

Locations

• French digital mathematics library (Numdam) - View - PDF
• Annales de l’institut Fourier - View - PDF