When does a right-angled Artin group split over $\mathbb{Z}$?

TL;DR

This paper characterizes when right-angled Artin groups split over infinite cyclic subgroups based on the connectivity of their defining graphs and computes their JSJ-decompositions in certain cases.

Contribution

It provides a graph-theoretic criterion for splitting over Z and explicitly computes JSJ-decompositions for 1-ended right-angled Artin groups.

Findings

A right-angled Artin group with at least three vertices does not split over Z iff its defining graph is biconnected.

The paper computes JSJ-decompositions for 1-ended right-angled Artin groups over infinite cyclic subgroups.

The results connect graph connectivity with algebraic splitting properties.

Abstract

We show that a right-angled Artin group, defined by a graph $\Gamma$ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if $\Gamma$ is biconnected. Further, we compute JSJ--decompositions of 1--ended right-angled Artin groups over infinite cyclic subgroups.