The topology of balls and Gromov hyperbolicity of Riemann surfaces

Details The topology of balls and Gromov hyperbolicity of Riemann surfaces The topology of balls and Gromov hyperbolicity of Riemann surfaces

2010-09-20

arxiv.org/abs/1009.3881v1

Mathematics

Differential Geometry

25 pages, 11 figures

Scientific paper

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.

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