Artin groups of type B and D

Abstract

We show that each of the Artin groups of type $B_n$ and $D_n$ can be presented as a semidirect product $F \rtimes {\cal B}_n$, where $F$ is a free group and ${\cal B}_n$ 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 ${\cal B}_n$ 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, ${\rm Out}(A(B_n),F)$ has order 2, and ${\rm Out}(A(D_n),F)$ 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 some orbifold. Applying the same techniques, we show that this latter group has an outer automorphism group of order 2. Finally, we determine the automorphism groups of all Artin groups or rank 2.