Right-angled Artin 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, for graphs with no edges, and free abelian groups $\Z^n$, for complete graphs. They are the Artin groups in which every relation is a commutation. They are CAT(0) groups, and they play a central role in the study of 3-manifolds and of special cube complexes.