Riemann surfaces have Hall rays at each cusp
Let \(\Sigma\) be a Riemann surface, \(\Sigma= \Gamma\setminus \mathbb{H}\), where \(\mathbb{H}\) denotes the upper half-plane and \(\Gamma\) a Fuchsian group. If \(p\) is a cusp of \(\Sigma\), the authors say that \(\Sigma\), the authors say that \(\Sigma\) has a Hall ray at \(p\) if there exists an \(N\) depending only on \(p\) and \(\Sigma\) such that there is a subset of the set of all heights of geodesics on \(\Sigma\) which is dense in \([N,\infty)\). The main result of the paper under review asserts that every Riemann surface admits a Hall ray at each cusp. For the modular surface this result is known and follows from Hall's theorem for continued fractions. Considering a suitable limit of cusped surfaces the authors show that a Hall ray can exist also without the presence of cusps. The limit procedure is applied to certain Hecke triangle surfaces and also produces an infinite class of closed geodesics on the theta surface that are pairwise equal in length but whose precursons in the limit process never are of equal length. In analogy with the parabolic case, hyperbolic Hall rays are constructed in the last section.
scientific article; zbMATH DE number 1043154