Artin groups of type B and D

摘要

We show that each of the Artin groups of type Bn and Dn can be presented as a semidirect product F x 38n, where F is a free group and 38" 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 38 " 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(Bn),F) has order 2, and Out(A(Dn), F) has order 4 if n is even and 2 if n is odd. It is known that the Artin group of type Dn may be viewed as an index 2 subgroup of the ^-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.