The topology of balls and Gromov hyperbolicity of Riemann surfaces

We prove that every ball in any non-exceptional Riemann surface with radius less or equal than 1\2 log3 is either simply or doubly connected. We use this theorem in order to study the hyperbolicity in the Gromov sense of Riemann surfaces. The results clarify the role of punctures and funnels of a Riemann surface in its hyperbolicity.