Artin–Tits group

Artin–Tits monoids are eligible for Garside methods based on the investigation of their divisibility relations, and are well understood:

Very few results are known for general Artin–Tits groups. In particular, the following basic questions remain open in the general case:

– determining the center — which is conjectured to be trivial or monogenic in the case when the group is not a direct product ("irreducible case"),

Partial results involving particular subfamilies are gathered below. Among the few known general results, one can mention:

Several important classes of Artin groups can be defined in terms of the properties of the Coxeter matrix.

Many other families of Artin–Tits groups have been identified and investigated. Here we mention two of them.