In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations.
They are closely related with Coxeter groups.
Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others.
The groups are named after Emil Artin, due to his early work on braid groups in the 1920s to 1940s,[1] and Jacques Tits who developed the theory of a more general class of groups in the 1960s.
is a (usually finite) set of generators and
is a set of Artin–Tits relations, namely relations of the form
, where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators
Alternatively, an Artin–Tits group can be specified by the set of generators
, the natural number
that is the length of the words
is the relation connecting
By convention, one puts
Formally, if we define
to denote an alternating product of
— the Artin–Tits relations take the form The integers
can be organized into a symmetric matrix, known as the Coxeter matrix of the group.
is an Artin–Tits presentation of an Artin–Tits group
obtained by adding the relation
is a Coxeter group.
is a Coxeter group presented by reflections and the relations
are removed, the extension thus obtained is an Artin–Tits group.
For instance, the Coxeter group associated with the
-strand braid group is the symmetric group of all permutations of
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: 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.