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A Primer on Mapping Class Groups (PMS-49)$
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Benson Farb and Dan Margalit

Print publication date: 2011

Print ISBN-13: 9780691147949

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691147949.001.0001

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Curves, Surfaces, and Hyperbolic Geometry

Curves, Surfaces, and Hyperbolic Geometry

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

Benson Farb

Dan Margalit

Princeton University Press

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

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