Large covers and sharp resonances of hyperbolic surfaces

TL;DR

This paper studies the behavior of Laplacian resonances on hyperbolic surfaces under large degree covers, revealing new existence results of sharp resonances near the critical line using thermodynamical and representation theory methods.

Contribution

It introduces novel results on the existence of sharp non-trivial resonances close to the critical line for large covers of hyperbolic surfaces, including abelian and congruence subgroup covers.

Findings

Existence of sharp non-trivial resonances near the line Re(s)=δ_Γ

Results hold for large degree abelian covers

Results extend to infinite index congruence subgroups

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 $SL2(\mathbb{Z})$.