Artin–Tits group

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

${\Big \langle }x_{1},x_{2},\ldots ,x_{n}{\Big |}\langle x_{1},x_{2}\rangle ^{m_{1,2}}=\langle x_{2},x_{1}\rangle ^{m_{2,1}},\ldots ,\langle x_{n-1},x_{n}\rangle ^{m_{n-1,n}}=\langle x_{n},x_{n-1}\rangle ^{m_{n,n-1}}{\Big \rangle }$

where

$m_{i,j}=m_{j,i}\in \{2,3,\ldots ,\infty \}$.

For $m<\infty$, $\langle x_{i},x_{j}\rangle ^{m}$ denotes an alternating product of $x_{i}$ and $x_{j}$ of length $m$, beginning with $x_{i}$. For example,

$\langle x_{i},x_{j}\rangle ^{3}=x_{i}x_{j}x_{i}$

and

$\langle x_{i},x_{j}\rangle ^{4}=x_{i}x_{j}x_{i}x_{j}$.

If $m=\infty$, then there is (by convention) no relation for $x_{i}$ and $x_{j}$.

The integers $m_{i,j}$ 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 ${x_{i}}^{2}=1$.

Classes of Artin groups

Braid groups are examples of Artin groups, with Coxeter matrix $m_{i,i+1}=3$ and $m_{i,j}=2$ for $|i-j|>1.$ 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 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. 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

$x_{i}x_{j}=x_{j}x_{i}\quad$ whenever i and j are connected by an edge in $\Gamma .$

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. 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.

References

• Mladen Bestvina, Noel Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), no. 3, 445-470.
• Pierre Deligne, Les immeubles des groupes de tresses généralisés. Invent. Math. 17 (1972), 273-302.
• Egbert Brieskorn, Kyoji Saito, Artin-Gruppen und Coxeter-Gruppen. Invent. Math. 17 (1972), 245--271.
• RUTH CHARNEY, AN INTRODUCTION TO RIGHT-ANGLED ARTIN GROUPS