---
tag: 8QE86W4GQ6
kind: definition
name: Artin group
msc: ["20F36"]
defines: SAWBDBHE41
uses: [G5S1M9, 0JGEQXQ4AC, G6FND7X69J, 08MG7H8JW5]
sources:
  - url: https://en.wikipedia.org/wiki/Artin%E2%80%93Tits_group
---

## Definition

Let $I$ be a set and let $M = (m_{ij})_{i, j \in I}$ be a Coxeter matrix, as in the definition of a [[Coxeter group|G6FND7X69J]].
For letters $x, y$ and an integer $m \ge 2$, let $\Pi(x, y; m) = x y x y \cdots$ be the alternating word of length $m$ that begins with $x$.
The *Artin group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]
$$A(M) = \langle s_i \ (i \in I) \mid \Pi(s_i, s_j; m_{ij}) = \Pi(s_j, s_i; m_{ij}) \text{ for all } i \ne j \text{ with } m_{ij} < \infty \rangle.$$
A group is an *Artin group*, or *Artin–Tits group*, if it is [[isomorphic|08MG7H8JW5]] to $A(M)$ for some Coxeter matrix $M$.

## Remarks

For $m_{ij} = 2$ the relation is $s_i s_j = s_j s_i$, and for $m_{ij} = 3$ it is $s_i s_j s_i = s_j s_i s_j$.
The quotient of $A(M)$ by the relations $s_i^2 = 1$ is the Coxeter group $W(M)$.
$A(M)$ is of *finite type*, or spherical, when $W(M)$ is finite.

Example: the [[braid group|BNR4H5GCTP]] $B_n$ is the Artin group with $I = \set{1, \ldots, n-1}$, $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\abs{i - j} \ge 2$.
These are exactly Artin's relations $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ and $\sigma_i \sigma_j = \sigma_j \sigma_i$.
If all $m_{ij} = \infty$, there are no relations and $A(M)$ is the [[free group|0KTW26C80J]] on $I$; if all $m_{ij} = 2$ for $i \ne j$, then $A(M)$ is free abelian.

Non-example: the group $\Z/2$ is not an Artin group.
In the abelianization of $A(M)$, a relation with $m_{ij}$ even becomes trivial, and a relation with $m_{ij}$ odd becomes $s_i = s_j$.
So the abelianization of an Artin group is free abelian, but $\Z/2$ is abelian and not free abelian.