  For each k > 0 we find an explicit function f_k such that the topology of S
inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian
surface (compact or noncompact) with K\geq -k^2, every point p on the surface,
and every r. 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\'e metric) obtained by deleting from one original surface S any
uniformly separated union of continua and isolated points.
