Simple geodesics and a series constant over Teichmüller space
type: Item
instance of: scholarly article
title: Simple geodesics and a series constant over Teichmüller space
zbMATH Open document ID: 0916.30039
published in: Inventiones Mathematicae
publication date: 1999-07-20T00:00:00Z
review text: Let \(M\) be a finite area Riemann surface with one cusp and no ramification. This paper offers the ramification following the spectacular identity: \(\sum{1\over 1+ \exp {1\over 2} (|\alpha |+ |\beta |)}= {1\over 2}\), where the sum is over all pairs of simple closed geodesics \(\alpha,\beta\) which bound an embedded pair of pants containing the cusp and \(|\;|\) gives the length. Additionally, the paper characterizes the \textit{Birman Series set} BS (the set of all points on all simple geodesics on \(M\) (which now may have more than one cusp)) near a fixed cusp. \textit{Near} means a horocyclic region \(H\) small enough so that any simple geodesic entering it must not leave; must terminate \textit{at} the cusp. The characterization is a bit too involved to state here, but it is given in terms of the isolated and boundary points of \(BS\cap \partial H\); which are themselves specified succintly and geometrically.
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 1186153
zbMATH Keywords: lamination
zbMATH Keywords: pair of pants
author: Greg McShane
reviewed by: Mark Sheingorn
MaRDI profile type: Publication
full work available at URL: s002220050235
OpenAlex ID: W2024202664
DOI: 10.1007/S002220050235
description: scientific article; zbMATH DE number 1186153