Every mapping class group is generated by 6 involutions

Abstract

Let Mod g, b denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Torsion elements in the mapping class group of a surface, math.GT/0004048, v1 8Apr2000] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod g, b . We answer Luo's question by proving that 3 torsion elements suffice to generate Mod g,0 . We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod g, b but also for the order of those elements. In particular, our main result is that 6 involutions (i.e., orientation-preserving diffeomorphisms of order two) suffice to generate Mod g, b for every genus g⩾3, b=0 and g⩾4, b=1.