Automorphism Groups of Some Affine and Finite Type Artin Groups

We observe that, for fixed n ≥ 3, each of the Artin groups of finite type An, Bn = Cn, and affine type Ãn−1 and C̃n−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. A rank n Coxeter matrix is a symmetric n× n matrix M with integer entries mij ∈ N∪ {∞} where mij ≥ 2 for i 6= j, and mii = 1 for all 1 ≤ i ≤ n. Given any rank n Coxeter matrix M , the Artin group of type M is defined by the presentation A(M) ∼= 〈 s1, . . . , sn | sisjsi . . . } {{ } mij = sjsisj . . . } {{ } mij for all i 6= j,mij 6= ∞〉 . Adding the relations si = 1 to this presentation yields a presentation of the Coxeter group of type M generated by standard reflections si and such that the rotation sisj has order mij, for all 1 ≤ i, j ≤ n. A Coxeter matrix M and its Artin group A(M) are said to be of finite type if the associated Coxeter group W (M) is finite, and of affine (or Euclidean) type if W (M) acts as a proper, cocompact group of isometries on some Euclidean space with the generators s1, . . . , sn acting as affine reflections. The information contained in the Coxeter matrix M is often displayed in the form of a graph, the Coxeter graph, whose vertices are numbered 1, .., n and which has an edge labelled mij between the vertices i and j whenever mij ≥ 3 or ∞. With this particular convention, one usually suppresses the labels which are equal to 3 (but not the corresponding edges!). Note that the absence of an edge between two vertices indicates that the corresponding generators of A(M) commute. We say that an Artin group is irreducible if its Coxeter graph is connected and observe that every Artin group is isomorphic to a direct product of irreducible Artin groups corresponding to the connected components of its Coxeter graph. In this paper we concern ourselves with the following four infinite families of Artin groups A = A(M) (see Figure 1): the finite typesM = An, Bn and affine types Ãn−1 and C̃n−1, of rank n ≥ 3 in each case. (We refer to [3] for the classification of irreducible finite and affine type Coxeter systems.) For each Artin group A on this list, we determine its automorphism group Aut(A), its outer automorphism group Out(A) = Aut(A)/Inn(A), and the abstract commensurator group Comm(A/Z) of the group modulo its centre (see Definition 3). The Artin group A(An) is well-known as the braid group on n + 1 strings, and the automorphism groups of the braid groups were determined by Dyer and Grossman in [7]. In this Date: August 26, 2004 . Charney was partially supported by NSF grant DMS-0104026.