Jump to ContentJump to Main Navigation
A Primer on Mapping Class Groups (PMS-49)$
Users without a subscription are not able to see the full content.

Benson Farb and Dan Margalit

Print publication date: 2011

Print ISBN-13: 9780691147949

Published to University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691147949.001.0001

Show Summary Details
Page of
date: 13 December 2017

Curves, Surfaces, and Hyperbolic Geometry

Curves, Surfaces, and Hyperbolic Geometry

Chapter:
(p.17) Chapter One Curves, Surfaces, and Hyperbolic Geometry
Source:
A Primer on Mapping Class Groups (PMS-49)
Author(s):

Benson Farb

Dan Margalit

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691147949.003.0002

This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a useful tool since each homotopy class of simple closed curves has a unique geodesic representative. The chapter begins by recalling some basic results about surfaces and hyperbolic geometry, with particular emphasis on the boundary of the hyperbolic plane and hyperbolic surfaces. It then considers simple closed curves in a surface S, along with geodesics and intersection numbers. It also discusses the bigon criterion, homotopy versus isotopy for simple closed curves, and arcs. Finally, it describes the change of coordinates principle and three facts about homeomorphisms.

Keywords:   simple closed curve, hyperbolic geometry, homotopy, hyperbolic plane, hyperbolic surface, intersection number, bigon criterion, isotopy, coordinates principle, homeomorphism

University Press Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs, and if you can't find the answer there, please contact us .