Gromov hyperbolicity of Riemann surfaces

20 pages, no figures.-- MSC2000 codes: 30F, 30F20, 30F45.MR#: MR2286916 (2007k:30080)Zbl#: Zbl 1115.30050In this paper we study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its "building block components". We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.The first author’s research is partially supported by a grant from DGI (BFM 2003-04870), Spain. The second author’s research is partially supported by a grant from DGI (BFM 2000-0022), Spain.Publicad