For finite denotes an alternating product of and of length , beginning with . For example,
If then there is (by convention) no relation for and .
The integers 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, known as the pure Artin group, is generated by relations of the form
Classes of Artin groups
Braid groups are examples of Artin groups, with Coxeter matrix and for Several important classes of Artin groups can be defined in terms of the properties of the 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 as well as Egbert Brieskorn and Kyoji 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 on n vertices labeled 1, 2, …, n defines a matrix M, for which mi,j = 2 if i and j are connected by an edge in and mi,j = ∞ otherwise. The right-angled Artin group associated with the matrix M has n generators x1, x2, …, xn and relations
whenever i and j are connected by an edge in
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 Joel Brady used right-angled Artin groups and their Salvetti complexes to construct groups with given finiteness properties.
Other Artin Groups
Kenneth Appel and Paul 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 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 freely generates a free subgroup of G.
- Theorem 4. An Artin or Coxeter group of extra-large type has solvable conjugacy problem.
- Free partially commutative monoid
- Artinian group (an unrelated notion)
- Non-commutative cryptography
- Elementary abelian group
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- Hermiller, Susan; Meier, John, Algorithms and geometry for graph products of groups (PDF)