Large covers and sharp resonances of hyperbolic surfaces

Abstract

Let $\Gamma$ be a convex co-compact discrete group of isometries of the hyperbolic plane ${\mathbb{H}}^2$, and $X=\Gamma \backslash {\mathbb{H}}^2$ 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 $\Gamma$. Using various techniques of thermodynamical formalism and representation theory, we prove two new existence results of "sharp non-trivial resonances" close to $\Re (s)={\delta }_{\Gamma }$, both in the large degree limit, for abelian covers and also infinite index congruence subgroups of $SL2(\mathbb{Z})$.

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

Cited by in corpus (1)

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