Rigidity Phenomena in the Mapping Class Group

Throughout this article we will consider connected orientable surfaces of negative Euler characteristic and of finite topological type, meaning of finite genus and with finitely many boundary components and/or cusps. We will feel free to think about cusps as marked points, punctures or topological ends. Sometimes we will need to make explicit mention of the genus and number of punctures of a surface: in this case, we will write Sg,n for the surface of genus g with n punctures and empty boundary. Finally, we define the complexity of a surface X as the number κ(X) = 3g − 3 + p, where g is the genus and p is the number of cusps and boundary components of X. In order to avoid too cumbersome notation, we denote by