Resonances on Some Geometrically Finite Hyperbolic Manifolds

Abstract

We first prove the meromorphic extension to ℂ for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the quotients Γ\ ⁿ⁺¹ with rational nonmaximal rank cusps previously studied by Froese-Hislop-Perry.

Keywords:

• Cusps
• Hyperbolic manifolds
• Resolvent
• Resonances

Mathematics Subject Classification:

• Primary 58J50
• Secondary 35P25

Acknowledgement

This work has been begun at Nantes University and finished at Purdue University. I would like to thank Peter Perry, Rafe Mazzeo and Martin Olbrich for pointing out to me some references about the subject.