Artin groups of type B and D

We show that each of the Artin groups of type B n and D n can be presented as a semidirect product F ⋊ ℬ n , where F is a free group and ℬ n is the n -string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ℬ n on the free group F is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup F is small: namely, Out( A ( B n ), F ) has order 2, and Out( A ( D n ), F ) has order 4 if n is even and 2 if n is odd. It is known that the Artin group of type D n may be viewed as an index 2 subgroup of the n -string braid group over a disk with a degree 2 orbifold point. We show that this orbifold braid group has outer automorphism group of order 2, for all n ≥ 2.