---
tag: SAWBDBHE41
kind: concept
name: Artin group
msc: ["20F36"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Artin%E2%80%93Tits_group
same_as:
  wikidata: Q3956993
  wikipedia: Artin–Tits group
---

## Summary

An Artin group, or Artin–Tits group, is a [[group|G5S1M9]] defined by a Coxeter matrix.
Each pair of generators satisfies a braid relation of a given length, or no relation.
Adding the relations $s_i^2 = 1$ turns it into the corresponding [[Coxeter group|G6FND7X69J]].
The [[braid groups|BNR4H5GCTP]], the [[free groups|0KTW26C80J]], the free abelian groups and the [[right-angled Artin groups|6D1DV09D6P]] are Artin groups.
Many basic questions about general Artin groups, such as whether they are torsion-free, are still open.

## Explanations

### L1 human

A braid relation says that two generators can be pushed past each other in a fixed pattern.
For example, $aba = bab$ is the relation of two neighbouring crossings in a braid with three strands.
An Artin group chooses, for each pair of generators, how long the alternating words in such a relation are, or imposes no relation at all.
It is like a Coxeter group without the rule that each generator squares to the identity, so the generators have infinite order.