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

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.