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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 Geometry

Teichmuller Geometry

Chapter:
(p.294) Chapter Eleven Teichmüller Geometry
Source:
A Primer on Mapping Class Groups (PMS-49)
Author(s):

Benson Farb

Dan Margalit

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691147949.003.0012

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

Keywords:   quasiconformal map, Teichmüller space, Teichmüller mapping, Teichmüller metric, metric geometry, existence theorem, holomorphic quadratic differential, measured foliations, Grötzsch's problem

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