Virtual splittings of right-angled Artin groups

Abstract

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

13 pages, 3 figures. Comments are welcome!