An approximate spectral representation and explicit bounds on Green functions of Fuchsian groups

TL;DR

This paper develops an approximate spectral representation for the Green function of the Laplace operator on hyperbolic surfaces formed by Fuchsian groups, providing explicit bounds using spectral and geometric analysis.

Contribution

It introduces a novel approximate spectral representation for the Green function and derives explicit bounds by combining spectral theory with geometric estimates.

Findings

Derived explicit bounds on the Green function for Fuchsian groups.

Established a limiting procedure from the resolvent kernel to the Green function.

Provided new estimates on Maa{f}ss forms and Eisenstein series.

Abstract

We study the Green function gr_\Gamma\ for the Laplace operator on the quotient of the hyperbolic plane by a cofinite Fuchsian group \Gamma. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of \Gamma\ on the hyperbolic plane to prove an "approximate spectral representation" for gr_\Gamma. Combining this with bounds on Maa{\ss} forms and Eisenstein series for \Gamma, we prove explicit bounds on gr_\Gamma.