---
tag: 6D1DV09D6P
kind: concept
name: Right-angled Artin group
msc: [20F36, 20F65]
specializes: [SAWBDBHE41]
sources:
  - url: https://en.wikipedia.org/wiki/Artin–Tits_group
---

## Summary

A right-angled Artin group is a group given by a finite simple graph, with one generator for each vertex.
Two generators commute whenever their vertices are joined by an edge, and there are no other relations.
These groups interpolate between [[free groups|0KTW26C80J]], for graphs with no edges, and free abelian groups $\Z^n$, for complete graphs.
They are the [[Artin groups|SAWBDBHE41]] in which every relation is a commutation.
They are [[CAT(0) groups|XB9QTY77ZA]], and they play a central role in the study of 3-manifolds and of special cube complexes.

## Explanations

### L1 human

Choose some letters, and decide for each pair whether the two letters may be swapped when they stand next to each other in a word.
Draw a graph with one vertex per letter and an edge for each swappable pair.
The right-angled Artin group of this graph is the group of words up to these swaps and cancellations of a letter with its inverse.
If no letters commute you get a free group; if all of them commute you get $\Z^n$.
For a path $a - b - c$, the letter $b$ commutes with everything, and the group is $F_2 \times \Z$.