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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Three

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Dehn Twists

In this chapter we study a particular type of mapping class called a Dehn twist. Dehn twists are the simplest infinite-order mapping classes in the sense that they have representatives with the smallest possible supports. Dehn twists play the role for mapping class groups that elementary matrices play for linear groups. We begin by defining Dehn twists in S and proving that they have infinite order in Mod(S). We determine many of the basic properties of Dehn twists by studying their action on simple closed curves. As one consequence, we compute the center of Mod(S). At the end of the chapter, we determine all relations ...

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