# A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature

@inproceedings{Benjamini2002ANA, title={A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature}, author={Itai Benjamini and Sergei Merenkov and Oded Schramm}, year={2002} }

Consider a simply-connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic, and negative mean excess implies that the surface is hyperbolic. Teichmuller gave an example of a hyperbolic simply-connected Riemann surface whose mean excess is zero, disproving the first of these implications. We give an example of a simply-connected parabolic…

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