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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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Teichmuller Space

Teichmuller Space

Chapter:
(p.263) Chapter Ten Teichmüller Space
Source:
A Primer on Mapping Class Groups (PMS-49)
Author(s):

Benson Farb

Dan Margalit

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691147949.003.0011

This chapter deals with Teichmüller space Teich(S) of a surface S. It first defines Teichmüller space and a topology on Teich(S) before giving two heuristic counts of its dimension. It then describes explicit coordinates on Teich(Sɡ) coming from certain length and twist parameters for curves in a pair of pants decomposition of Sɡ; these are the Fenchel–Nielsen coordinates on Teich(Sɡ). The chapter also considers the Teichmüller space of the torus and concludes by proving the 9g – 9 theorem, which states that a hyperbolic structure on Sɡ is completely determined by the lengths assigned to 9g – 9 isotopy classes of simple closed curves in Sɡ.

Keywords:   Teichmüller space, topology, Fenchel–Nielsen coordinates, torus, hyperbolic structure, isotopy, simple closed curve

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