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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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The Nielsen-Thurston Classification

The Nielsen-Thurston Classification

Chapter:
(p.367) Chapter Thirteen The Nielsen-Thurston Classification
Source:
A Primer on Mapping Class Groups (PMS-49)
Author(s):

Benson Farb

Dan Margalit

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691147949.003.0014

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Keywords:   mapping class group, Nielsen–Thurston classification, torus, 3-manifold theory, geometric classification, mapping torus, collar lemma, hyperbolic geometry, surface

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