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

Peter Bruin

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arXiv · 1205.6306

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

Peter Bruin

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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.

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Peter Bruin. 2012-05-29. An approximate spectral representation and explicit bounds on Green functions of Fuchsian groups. https://arxiv.org/abs/1205.6306

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