Virtual splittings of right-angled Artin groups

Abstract

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

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop:ArtinSubAbelian']}.
• Definition 3.1
• Definition 3.2