Automorphism groups of some affine and finite type Artin groups

We observe that, for fixed $n\geq 3$, each of the Artin groups of finite type $A_n$, $B_n=C_n$, and affine type $\tilde A_{n-1}$ and $\tilde C_{n-1}$ is a central extension of a finite index subgroup of the mapping class group of the $(n+2)$-punctured sphere. (The centre is trivial in the affine case and infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz on abstract commensurators of surface mapping class groups we are able to determine the automorphism groups of each member of these four infinite families of Artin groups.

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Published 1 January 2005