In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form
Contents
- Classes of Artin groups
- Artin groups of finite type
- Right angled Artin groups
- Other Artin Groups
- References
where
For
and
If
The integers
Classes of Artin groups
Braid groups are examples of Artin groups, with Coxeter matrix
Artin groups of finite type
If M is a Coxeter matrix of finite type, so that the corresponding Coxeter group W = A(M) is finite, then the Artin group A = A(M) is called an Artin group of finite type. The 'irreducible types' are labeled as An , Bn = Cn , Dn , I2(n) , F4 , E6 , E7 , E8 , H3 , H4 . A pure Artin group of finite type can be realized as the fundamental group of the complement of a finite hyperplane arrangement in Cn. Pierre Deligne and Brieskorn-Saito have used this geometric description to compute the center of A, its cohomology, and to solve the word and conjugacy problems.
Right-angled Artin groups
If M is a matrix all of whose elements are equal to 2 or ∞, then the corresponding Artin group is called a right-angled Artin group, but also a (free) partially commutative group, graph group, trace group, semifree group or even locally free group. For this class of Artin groups, a different labeling scheme is commonly used. Any graph Γ on n vertices labeled 1, 2, …, n defines a matrix M, for which mij = 2 if i and j are connected by an edge in Γ, and mij = ∞ otherwise. The right-angled Artin group A(Γ) associated with the matrix M has n generators x1, x2, …, xn and relations
The class of right-angled Artin groups includes the free groups of finite rank, corresponding to a graph with no edges, and the finitely-generated free abelian groups, corresponding to a complete graph. In fact, every right-angled Artin group of rank r can be constructed as HNN extension of a right-angled Artin group of rank r − 1, with the free product and direct product as the extreme cases. A generalization of this construction is called a graph product of groups. A right-angled Artin group is a special case of this product, with every vertex/operand of the graph-product being a free group of rank one (the infinite cyclic group).
Mladen Bestvina and Noel Brady constructed a nonpositively curved cubical complex K whose fundamental group is a given right-angled Artin group A(Γ). They applied Morse-theoretic arguments to their geometric description of Artin groups and exhibited first known examples of groups with the property (FP2) that are not finitely presented.
Other Artin Groups
We define that an Artin group or a Coxeter group is of large type if mi j ≥ 3 for all i ≠ j. We say that an Artin group or a Coxeter group is of extra-large type if mi j ≥ 4 for all i ≠ j.
Kenneth Appel and P.E. Schupp looked further into Artin groups and the properties that hold true for them. They proved four theorems, which were known to be true for Coxeter groups, and showed that they also held for Artin groups. Appel and Schupp had discovered that they could study extra-large Artin and Coxeter groups through the techniques of small cancellation theory. They also discovered that they could use a "refinement" of these same techniques to work with these groups of large type.
Theorem 1 : Let G be an Artin or Coxeter group of extra-large type. If J ⊆ I then GJ has a presentation defined by the Coxeter matrix MJ and the generalized word problem for GJ in G is solvable. If J, K ⊆ I then GJ ∩ GK = G (J ∩ K).
Theorem 2 : An Artin group of extra-large type is torsion-free.
Theorem 3 : Let G be an Artin group of extra-large type. Then the set {ai2 : i ∈ I} freely generates a free subgroup of G.
Theorem 4 : An Artin or Coxeter group of extra-large type has solvable conjugacy problem.