Automorphism groups of some affine and finite type Artin groups

By Ruth Charney, J. Crisp. 2004

51 citations recorded by the source corpus.

We observe that, for fixed n ≥ 3, each of the Artin groups of finite type An, Bn = Cn, and affine type ˜ An−1 and ˜ Cn−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 ij, 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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/effbb86d5f7eaf3f74c5eed193ed815986549b85

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Citations in the corpus

An introduction to right-angled Artin groups (2006, unknown): “See [25] and [3] for a discussion of these groups.”
Asymptotic Dimension (2007, background): “In [30], Charney and Crisp observe that each of the Artin groups A(An), A(Bn) of finite type and the Artin groups A(Ãn−1) and A(C̃n−1) of affine”
The asymptotic dimension of a curve graph is finite (2005, background): “In [15], Charney and Crisp observe that each of the Artin groups A(An), A(Bn) of finite type and the Artin groups A(Ãn−1) and A(C̃n−1) of affine type is a central extension of a finite index subgroup of Mod(S0,n+2) when n ≥ 3.”
Injections of Artin groups (2005, unknown): “the proof of the corollary, it suffices to note that P (An)/Z is not a subgroup of A(Ãn−1) (see [9]) and that P (An)/Z, contained in each Gk, is normal in Mod(Sm).”
Curve Complexes and Finite Index Subgroups of Mapping Class Groups (2005, background): “When, in the conclusion of the proposition, is trivial, we remark that ρ is a transvection in a sense slightly generalizing that of Charney and Crisp [8] (see also [3, 24]).”