*Chapter Seven*

### 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(*S*_{g}). We will show that, for *g* ≥ 2, finite subgroups have order at most 84(*g* − 1) and cyclic subgroups have order at most 4*g* + 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(*S*_{g}) is generated by finitely many elements of order 2.

**7.1 FINITE-ORDER MAPPING CLASSES ...**