Torsion

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen realization theorem for cyclic groups and detection of torsion with the symplectic representation. It then considers the problem of finding an Euler characteristic for orbifolds, to prove a Gauss–Bonnet theorem for orbifolds, and to use these results to show that there is a universal lower bound of π‎/21 for the area of any 2-dimensional orientable hyperbolic orbifold. The chapter demonstrates that, when g is greater than or equal to 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. It also describes finitely many conjugacy classes of finite subgroups in Mod(S) and concludes by proving that Mod(Sɡ) is generated by finitely many elements of order 2.