On a theorem of Artin

Arjeh M. Cohen & Luis Paris

In 1947, Artin published the following two results on the braid group B_{n} on n strings.

Theorem 1. (Artin). Suppose that φ : B_{n} → Sym_{n} is an epimorphism. Then, up to an automorphism of Sym_{n}, the elements φ(σ_{i}) (i = 1, …, n−1) are the standard generators (i, i + 1) of Sym_{n}, with two exceptions for n = 4. Theorem 2. (Artin). The pure braid group P B_{n} is a characteristic subgroup of B_{n}.

Let Γ be a Coxeter graph with set of vertices {1, …, n}. For two objects a, b and for m ∈ N, define the word

[a, b⟩^{m} = \begin{cases}(ab)^{\frac{m}{2}} & if m is even, \\ (ab)^{\frac{m−1}{2}} a & if m is odd. \end{cases}

The Artin group of type Γ is the group A generated by n elements σ_{1}, …, σ_{n}, and subject to the relations

[σ_{i}, σ_{j}⟩^{m_{i,j}} = [σ_{j}, σ_{i}⟩^{m_{i,j}}

for 1 ≤ i < j ≤ n, and m_{i,j} ≠ ∞, where (m_{i,j})_{i,j} is the Coxeter matrix of Γ. The Coxeter system of type Γ is denoted by (W, S) with S consisting of the images s_{i} of σ_{i} (i = 1, …, n) under the natural epimorphism μ : A → W. We say that A is irreducible if Γ is connected, and that A is of spherical type if W is finite.

In this lecture we prove the results corresponding to Theorems 1 and 2 for the irreducible Artin groups of spherical type, and we explain some open questions related to them.