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

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Chapter Thirteen

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The Nielsen–Thurston Classification

In this chapter we explain and prove one of the central theorems in the study of mapping class groups: the Nielsen–Thurston classification of elements of Mod(S). This theorem is the analogue of the Jordan canonical form for matrices. It states that every f euro Mod(S) is one of three special types: periodic, reducible, or pseudo-Anosov. The knowledge of individual mapping classes is essential to our understanding of the algebraic structure of Mod(S). As we will soon explain, it is also essential for ...

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