Geodesic excursions to infinity in Riemann surfaces
Let \(S\) be a Riemann surface that has the unit disk \(\mathbb{D}\) as its universal covering surface: thus \(S\) is conformally equivalent to \(\mathbb{D}/G\) for some Fuchsian group \(G\). The authors prove a very interesting relation between the Hausdorff dimension of the geodesics that go to the infinity of \(S\) and the decay of the injectivity radius of \(S\). This can be expressed as follows: Let \(L_ c(G)\) be the conical (angular) limit set of the group \(G\). Then, for every \(\varepsilon> 0\), there exist \(r_ n\to 1\) and \(| z_ n|< r_ n\), \(g_ n\in G\), \(g_ n\neq id\) such that \[ d_{\mathbb{D}}(z_ n, g_ n(z_ n))< (1- r_ n)^{1-\dim\otimes \mathbb{D}\backslash L_ c(G))- \varepsilon}. \] The authors present two proofs. One uses a theorem of Littlewood about the growth of analytic functions with range in a given domain and works for the case that \(S\) is planar. The proof for the general cases uses, among other things, an idea of Sullivan. There are several remarks and examples.
scientific article; zbMATH DE number 718517