Virtual splittings of right-angled Artin groups

In this article, we determine, given a finite graph $\mathrm{\Gamma }$ and an integer $n\ge 1$, when a right-angled Artin group $A(\mathrm{\Gamma })$ virtually splits over an abelian subgroup of rank $n$. More precisely, we show that the following assertions are equivalent: (1) $A(\mathrm{\Gamma })$ admits ${\mathbb{Z}}^n$ as a codimension-one subgroup, (2) $A(\mathrm{\Gamma })$ virtually splits over ${\mathbb{Z}}^n$, (3) $A(\mathrm{\Gamma })$ splits over ${\mathbb{Z}}^n$, and (4) $\mathrm{\Gamma }$ either is a complete graph with $n+1$ vertices or contains a complete subgraph of size $n$ that has a subgraph separating $\mathrm{\Gamma }$.