Free products in mapping class groups generated by Dehn twists

Abstract

Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).