Simplicial structures and normal forms for mapping class groups and braid groups

Abstract

In this paper we show that the mapping class groups of any surface with nonempty boundary form a simplicial group as the number of marked points varies. This extends the simplicial structure on braid groups of surfaces found by Berrick, Cohen, Wong and Wu. We use the simplicial maps to construct compatible normal forms for elements of the braid groups and mapping class groups of surfaces with boundary.