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

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