Stable commutator length in the mapping class group

Ken Bromberg (Utah)

Abstract: The mapping class group is an example of a perfect group; its abelianization is trivial. In particular, every element can be written as a product of commutators. Endo and Kotschik showed that the mapping class group is not uniformly perfect; there is no bound on the number of commutators required to represent a given element. To prove this they showed that there are elements with positive "stable commutator length." Their proof uses rather sophisticated results on the symplectic geometry of 4-manifolds. In this talk we will use more elementary methods to give a complete characterization of when the stable commutator length is positive in the mapping class group. The is joint work with M. Bestvina and K. Fujiwara.