Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

TL;DR

This paper establishes the meromorphic extension of the Laplacian resolvent and related operators on geometrically finite hyperbolic manifolds, and analyzes lattice point asymptotics using the limit set's Hausdorff dimension.

Contribution

It extends the resolvent and scattering operator to the entire complex plane and links lattice point asymptotics to the limit set's Hausdorff dimension.

Findings

Meromorphic extension of the resolvent, Poincaré series, Eisenstein series, and scattering operator.

Asymptotic behavior of lattice points related to the Hausdorff dimension.

Unified analysis of spectral and geometric properties of hyperbolic manifolds.

Abstract

For geometrically finite hyperbolic manifolds $\Gamma\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincar\'e series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of $\Gamma$ in large balls of $H^{n+1}$ in terms of the Hausdorff dimension of the limit set of $\Gamma$.