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Abstract. In 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 to it. These results clarify how the decomposition of a Riemann surface in Y -pieces and funnels affects on the hyperbolicity of the surface. The results simplify the topology of the surface and allow to obtain global results from local information.

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.

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Abstract. In this paper we characterize the Gromov hyperbolicity of the double of a metric space. This result allows to give a characterization of the hyperbolic Denjoy domains, in terms of the distance to R of the points in some geodesics. In the particular case of trains (a kind of Riemann surfaces which includes the flute surfaces), we obtain more explicit criteria which depend just on the lengths of what we have called fundamental geodesics.