Riemann surfaces have Hall rays at each cusp
type: Item
instance of: scholarly article
title: Riemann surfaces have Hall rays at each cusp
zbMATH Open document ID: 0877.30019
author: Mark Sheingorn
author: Thomas A. Schmidt
published in: Illinois Journal of Mathematics
publication date: 1997-12-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 1043154
zbMATH Keywords: Riemann surface
zbMATH Keywords: Hall's theorem
zbMATH Keywords: Hall ray
zbMATH Keywords: Hecke triangle surface
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1043154