Resonances and convex co-compact congruence subgroups of PSL2(Z)

TL;DR

This paper investigates the resonance spectrum of convex co-compact congruence subgroups of PSL2(Z) as the congruence parameter increases, revealing contrasting behaviors in different regions of the complex plane.

Contribution

It provides new bounds on the resonance spectrum for these subgroups, highlighting distinct behaviors across the critical line.

Findings

Lower bounds for the resonance counting function in discs.

Upper bounds for the resonance counting function in vertical strips.

Different spectral behaviors on either side of the critical line =δ/2.

Abstract

This papers deals with congruence subgroups of convex cocompact subgroups of PSL2(Z). We examine the behaviour of the resonance spectrum when the congruence parameter q goes to infinity: we show a lower bound for the counting function in discs and an upper bound in vertical strips. These results show drastically different behaviour on both sides of the critical line $\Re(s)=\delta/2$.