Artin group

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

[math]\displaystyle{ \begin{align} \Big\langle x_1,x_2,\ldots,x_n \Big| \langle x_1, x_2 \rangle^{m_{1,2}} & =\langle x_2, x_1 \rangle^{m_{2,1}}, \ldots \\ & \ldots, \langle x_{n-1}, x_n \rangle^{m_{n-1,n}} = \langle x_n, x_{n-1} \rangle^{m_{n,n-1}} \Big\rangle \end{align} }[/math]

where

[math]\displaystyle{ m_{i,j} = m_{j,i} \in \{2,3,\ldots, \infty\}. }[/math]

For [math]\displaystyle{ m \lt \infty }[/math], [math]\displaystyle{ \langle x_i, x_j \rangle^m }[/math] denotes an alternating product of [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ x_j }[/math] of length [math]\displaystyle{ m }[/math], beginning with [math]\displaystyle{ x_i }[/math]. For example,

[math]\displaystyle{ \langle x_i, x_j \rangle^3 = x_ix_jx_i }[/math]

and

[math]\displaystyle{ \langle x_i, x_j \rangle^4 = x_ix_jx_ix_j. }[/math]

If [math]\displaystyle{ m=\infty }[/math], then there is (by convention) no relation for [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ x_j }[/math].

The integers [math]\displaystyle{ m_{i,j} }[/math] 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 [math]\displaystyle{ {x_i}^2=1 }[/math].

Classes of Artin groups

Braid groups are examples of Artin groups, with Coxeter matrix [math]\displaystyle{ m_{i,i+1} = 3 }[/math] and [math]\displaystyle{ m_{i,j}=2 }[/math] for [math]\displaystyle{ |i-j|\gt 1. }[/math] 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 A_n , B_n = C_n , D_n , 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 Γ on n vertices labeled 1, 2, …, n defines a matrix M, for which m_ij = 2 if i and j are connected by an edge in Γ, and m_ij = ∞ otherwise. The right-angled Artin group A(Γ) associated with the matrix M has n generators x₁, x₂, …, x_n and relations

[math]\displaystyle{ x_i x_j = x_j x_i \quad }[/math] whenever i and j are connected by an edge in [math]\displaystyle{ \Gamma. }[/math]

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 m_{i j} ≥ 3 for all i ≠ j. We say that an Artin group or a Coxeter group is of extra-large type if m_{i j} ≥ 4 for all i ≠ j.

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 these groups of large type.

Theorem 1 : Let G be an Artin or Coxeter group of extra-large type. If J ⊆ I then G_J has a presentation defined by the Coxeter matrix M_J and the generalized word problem for G_J in G is solvable. If J, K ⊆ I then G_J ∩ G_K = 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 {a_i² : i ∈ I} freely generates a free subgroup of G.

Theorem 4 : An Artin or Coxeter group of extra-large type has solvable conjugacy problem.

See also

• Free partially commutative monoid
• Artinian group (an unrelated notion)
• Non-commutative cryptography
• Elementary abelian group

References

• Bestvina, Mladen; Brady, Noel (1997), "Morse theory and finiteness properties of groups", Inventiones Mathematicae 129 (3): 445–470, doi:10.1007/s002220050168, Bibcode: 1997InMat.129..445B, https://link.springer.com/article/10.1007/s002220050168 
• Deligne, Pierre (1972), "Les immeubles des groupes de tresses généralisés", Inventiones Mathematicae 17 (4): 273–302, doi:10.1007/BF01406235, Bibcode: 1972InMat..17..245B, https://link.springer.com/article/10.1007/BF01406235 
• Brieskorn, Egbert; Saito, Kyoji (1972), "Artin-Gruppen und Coxeter-Gruppen", Inventiones Mathematicae 17 (4): 245–271, doi:10.1007/bf01406235, Bibcode: 1972InMat..17..245B 
• Charney, Ruth (2007), "An introduction to right-angled Artin groups", Geometriae Dedicata 125 (1), doi:10.1007/s10711-007-9148-6 
• Casals-Ruiz, Montserrat; Kazachkov, Ilya (2009), On systems of equations over free partially commutative groups, Bibcode: 2008arXiv0810.4867C 
• Esyp, Evegenii S.; Kazachkov, Ilia V.; Remeslennikov, Vladimir N. (2005), Divisibility theory and complexity of algorithms in free partially commutative groups, Contemporary Mathematics, 378, Providence, RI: American Mathematical Society, Bibcode: 2005math.....12401E 
• Hermiller, Susan; Meier, John (PDF), Algorithms and geometry for graph products of groups, http://www.math.unl.edu/~shermiller2/webppr/graphproduct.pdf