The topology of balls in Riemannian surfaces and Gromov hyperbolicity

Abstract

For each \(k>0\) we find an explicit function \(f_k\) such that the topology of \(S\) inside the ball \(B_S(p,r)\) is ‘bounded’ by \(f_k(r)\) for every complete Riemannian surface (compact or non-compact) \(S\) with \(K \ge -k^2\), every \(p \in S\) 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 Poincare metric) obtained by deleting from one original surface \(S\) any uniformly separated union of continua and isolated points.