  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.
