Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

Abstract

On geometrically finite hyperbolic manifolds $\Gamma \backslash {\mathbb{H}}^d$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $\Gamma$ satisfy a certain Diophantine condition, the bound is $N(R)=\mathcal{O}(R^d(\log R{)}^{d+1})$.