Right-angled Artin group

Remarks

Example: if $\Gamma$ has $n$ vertices and no edges, $A(\Gamma)$ is the free group $F_n$; if $\Gamma$ is complete, $A(\Gamma)$ is the free abelian group $\Z^n$. Example: if $\Gamma$ is the path $a - b - c$, the relations say that $b$ commutes with $a$ and $c$, and $a, c$ generate a free group, so $A(\Gamma) \cong F_2 \times \Z$.

Non-example: the braid group $B_3 = \langle a, b \mid aba = bab \rangle$ is an Artin group but not a right-angled one. Its abelianisation is $\Z$, since the relation forces $a = b$ there; the only right-angled Artin group with abelianisation $\Z$ is $\Z$ itself, and $B_3$ is not abelian.

The abelianisation of $A(\Gamma)$ is $\Z^{V}$. Two right-angled Artin groups $A(\Gamma)$ and $A(\Gamma')$ are isomorphic if and only if the graphs $\Gamma$ and $\Gamma'$ are isomorphic. They are also called graph groups or partially commutative groups.