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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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Dehn Twists

Dehn Twists

(p.64) Chapter Three Dehn Twists
A Primer on Mapping Class Groups (PMS-49)

Benson Farb

Dan Margalit

Princeton University Press

This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

Keywords:   mapping class group, Dehn twists, simple closed curve, geometric intersection number, algebraic relations, surface, geometric operation, inclusion homomorphism, capping homomorphism, cutting homomorphism

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