Artin group

Summary

An Artin group, or Artin–Tits group, is a group defined by a Coxeter matrix. Each pair of generators satisfies a braid relation of a given length, or no relation. Adding the relations $s_i^2 = 1$ turns it into the corresponding Coxeter group. The braid groups, the free groups, the free abelian groups and the right-angled Artin groups are Artin groups. Many basic questions about general Artin groups, such as whether they are torsion-free, are still open.