Geodesics on quotient manifolds and their corresponding limit points
type: Item
instance of: scholarly article
title: Geodesics on quotient manifolds and their corresponding limit points
zbMATH Open document ID: 1044.37018
author: Torbjörn Lundh
published in: Michigan Mathematical Journal
publication date: 2003-10-27T00:00:00Z
review text: Let \(\Gamma\) be a Fuchsian or Kleinian group acting on the unit circle or ball \(B\); let \(S\) be the unit circle or sphere, i.e., the boundary of \(B\). Let \(L\) be the limit set of \(\Gamma\). The author considers half-geodesics, i.e., maps \(\gamma:[0,\infty[ \rightarrow B\) which are geodesic. This defines a point \(\xi(\gamma)\in S\). Let \(\phi_\gamma(t)\) be the distance, in \(\Gamma \backslash B\) between \(\gamma(0)\) and \(\gamma(t)\). If \(\phi_\gamma\) is bounded then \(\xi(\gamma)\) lies in the conical limit set of \(\Gamma\). The main theorem of this paper describes the geometrical nature of the limit point \(\xi(\gamma)\) in terms of the asymptotics \(\phi_\gamma\). The proof uses the methods of hyperbolic geometry. The paper concludes with some related results and examples.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 1997313
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Kleinian group
zbMATH Keywords: limit set
zbMATH Keywords: conical limit point
zbMATH Keywords: geodesic flow
zbMATH Keywords: escape rate
zbMATH Keywords: archipelago
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1060013197
OpenAlex ID: W2059555371
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DOI: 10.1307/MMJ/1060013197
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description: scientific article; zbMATH DE number 1997313