The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces

27 pages, no figures.-- MSC2000 codes: 30F20, 30F45.MR#: MR2243795 (2007e:30063)Zbl#: Zbl 1108.30031We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface S* obtained by deleting a closed set from one original surface S. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.Research by all three authors partially supported by a grant from DGI (BFM 2003-04870), Spain. In addition, research by third author (Eva Tourís) was partially supported by a grant from DGI (BFM 2000-0022), Spain.Publicad