Linearity of some low-complexity mapping class groups

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus g with b boundary components and n punctures in the cases when g = 0 with b and n arbitrary, and when g = 1 and b + n is at most 3, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type D-4. As a corollary, we conclude that the pure mapping class groups are linear in these cases.