Lower bounds for resonances of infinite-area Riemann surfaces

Abstract

For infinite-area, geometrically finite surfaces $X=\Gamma \setminus {ℍ}^2$, we prove new omega lower bounds on the local density of resonances $\mathcal{D}\left(z\right)$ when $z$ lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension $\delta$ of the limit set of $\Gamma$. The first bound is valid when $\delta >\frac{1}{2}$ and shows logarithmic growth of the number $\mathcal{D}\left(z\right)$ of resonances at high energy, that is, when $|Re\left(z\right)|\to +\infty$. The second bound holds for $\delta >\frac{3}{4}$ and if $\Gamma$ is an infinite-index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillopé and Zworski on the existence of a fractal Weyl law for resonances.