Virtual splittings of right-angled Artin groups

TL;DR

This paper characterizes when right-angled Artin groups virtually split over abelian subgroups of a given rank based on properties of their defining graphs.

Contribution

It provides a complete characterization linking graph properties to the virtual splitting over abelian subgroups in right-angled Artin groups.

Findings

Equivalent conditions for virtual splitting over Z^n are established.

Splitting over Z^n occurs if and only if the graph contains specific complete subgraphs.

The results connect graph structure directly to algebraic splitting properties.

Abstract

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