Minimal generating sets for the mapping class group

Abstract

The mapping class group of an orientable surface is the group of orientation-preserving self-diffeomorphisms of the surface modulo isotopy. This family of groups plays a central role in low dimensional topology. Therefore, its algebraic structure is of interest. This chapter focuses on the generating sets of the mapping class groups, specifically various minimal generating sets. After the necessary definitions are given, Dehn twists and relations among them are introduced. Next, the minimal generating sets of the mapping class group are considered: It turns out that mapping class group is generated by two elements, by two torsion elements, and by four involutions. Finally, known results about the minimal generating sets for the extended mapping class groups, the hyperelliptic mapping class groups, and the mapping class groups of nonorientable surface are considered.