Artin Relations in the Mapping Class Group

Abstract

For every integer l bigger than one, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length l. That is, xyx... = yxy..., where each side of the equality contains l terms. By direct computations, we first find elements x and y in Mod(S) satisfying Artin relations of every even length bigger than 6, and every odd length bigger than 1. Then using the theory of Artin groups, we give two more alternative ways for finding Artin relations in Mod(S). The first provides Artin relations of every length greater than 3, while the second produces Artin relations of every even length greater than 4.