Artin group

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In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

\left\langle x_1,x_2,\ldots,x_n|\langle x_1, x_2 \rangle^{m_{1,2}}=\langle x_2, x_1 \rangle^{m_{1,2}}, \langle x_1, x_3 \rangle^{m_{1,3}}=\langle x_3, x_1 \rangle^{m_{1,3}}, \ldots \langle x_{n-1}, x_n \rangle^{m_{n-1,n}}=\langle x_{n-1}, x_n \rangle^{m_{n,n-1}}\right\rangle

where

m_{i,j} \in {2,3,\ldots, \infty}.

For m < \infty, \langle x_i, x_j \rangle^m represents an alternating product of xi and xj of length m, beginning with xi. (For example, \langle x_i, x_j \rangle^3 = x_ix_jx_i and \langle x_i, x_j \rangle^4 = x_ix_jx_ix_j.) If m=\infty, then there is no relation for xi and xj.

The mi,j can be organized into a symmetric matrix, known as the Coxeter matrix of the group. Each Artin group has as a quotient the Coxeter group with the same set of generators and Coxeter matrix. The kernel of the homomorphism to the associated Coxeter group is generated by relations of the form {x_i}^2=1.

Braid groups are examples of Artin groups, with Coxeter matrix mi,i + 1 = 3 and mi,j = 2 for | i − j | > 1.

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