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

### David Borthwick

Emory University, Atlanta, USA### Colin Guillarmou

Ecole Normale Superieure, Paris, France

## 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 $N(R)=\mathcal O(R^d (\mathrm {log} R)^{d+1})$.

## Cite this article

David Borthwick, Colin Guillarmou, Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds. J. Eur. Math. Soc. 18 (2016), no. 5, pp. 997–1041

DOI 10.4171/JEMS/607