The topology of balls in Riemannian surfaces and Gromov hyperbolicity

For each k > 0 we find an explicit function fk such that the topology of S inside the ball BS (p,r) is 'bounded' by fk (r) for every complete Riemannian surface (compact or non-compact) with S K >-k2 , every p epsilon S and every r > 0. Using this result, we obtain a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a Riemann surface S* (with its own Poincar, metric) obtained by deleting from one original surface S any uniformly separated union of continua and isolated points.

The topology of balls in Riemannian surfaces and Gromov hyperbolicity Articles