Coarse separation and splittings in right-angled Artin groups

TL;DR

This paper characterizes when right-angled Artin groups split over abelian subgroups, linking geometric splitting properties to graph-theoretic conditions involving completeness and separation.

Contribution

It provides a precise geometric criterion for splitting over abelian subgroups based on the structure of the defining graph.

Findings

A right-angled Artin group splits over an abelian subgroup iff its defining graph is complete or separated by a complete subgraph.

Splitting over abelian subgroups corresponds to coarse separability by subexponential growth.

The characterization connects algebraic splittings with combinatorial properties of the defining graph.

Abstract

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $\Gamma$, we show that $A(\Gamma)$ splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that $\Gamma$ is complete or separated by a complete subgraph.