#### Vol. 3, No. 2, 2010

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Lower bounds for resonances of infinite-area Riemann surfaces

### Dmitry Jakobson and Frédéric Naud

Vol. 3 (2010), No. 2, 207–225
##### Abstract

For infinite-area, geometrically finite surfaces $X=\Gamma \setminus {ℍ}^{2}$, we prove new omega lower bounds on the local density of resonances $\mathsc{D}\left(z\right)$ 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 $\mathsc{D}\left(z\right)$ 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.

##### Keywords
Laplacian, resonances, arithmetic fuchsian groups
##### Mathematical Subject Classification 2000
Primary: 11F72, 58J50