Automorphism groups of some affine and finite type Artin groups. (English) Zbl 1077.20055

Charney, Ruth; Crisp, John

Math. Res. Lett. 12, No. 2-3, 321-333 (2005).

Summary: 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 \(\widetilde A_{n-1}\) and \(\widetilde C_{n-1}\) 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.

MSC:

20F36	Braid groups; Artin groups
20F28	Automorphism groups of groups
57M07	Topological methods in group theory

Keywords:

Artin groups of finite type; commensurators; surface mapping class groups; automorphism groups