Splittings of right-angled Artin groups

TL;DR

This paper proves that non-trivial actions of right-angled Artin groups on trees imply the existence of separating subgraphs in their defining graphs, revealing structural properties of these groups.

Contribution

It establishes a link between group actions on trees and the combinatorial structure of the defining graph, specifically identifying separating subgraphs.

Findings

Non-trivial minimal actions on trees imply the existence of separating subgraphs.

Separating subgraphs correspond to stabilizers of edges in the action.

Provides a criterion for understanding group actions via graph structure.

Abstract

We show that if a right-angled Artin group $A(\Gamma)$ has a non-trivial, minimal action on a tree $T$ which is not a line, then $\Gamma$ contains a separating subgraph $\Lambda$ such that $A(\Lambda)$ stabilizes an edge in $T$.