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

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Chapter Seven

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Torsion

In this chapter we investigate finite subgroups of the mapping class group. After explaining the distinction between finite-order mapping classes and finite-order homeomorphisms, we then turn to the problem of determining what is the maximal order of a finite subgroup of Mod(Sg). We will show that, for g ≥ 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. We will also see that there are finitely many conjugacy classes of finite subgroups in Mod(S). At the end of the chapter, we prove that Mod(Sg) is generated by finitely many elements of order 2.

7.1 FINITE-ORDER MAPPING CLASSES ...

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