Generalising some results about right-angled Artin groups to graph products of groups

TL;DR

This paper extends key results about right-angled Artin groups to the broader context of graph products of groups, providing new conditions for kernel freeness, group series structure, and hyperbolicity.

Contribution

It generalizes three fundamental results from right-angled Artin groups to graph products, offering new criteria and proofs for kernel properties, group decompositions, and hyperbolicity.

Findings

Characterizes when the kernel of the natural map is free

Shows existence of a free product series for groups with finite chromatic number

Provides an alternative proof for conditions of hyperbolicity

Abstract

We prove three results about the graph product $G=\G(\Gamma;G_v, v \in V(\Gamma))$ of groups $G_v$ over a graph $\Gamma$. The first result generalises a result of Servatius, Droms and Servatius, proved by them for right-angled Artin groups; we prove a necessary and sufficient condition on a finite graph $\Gamma$ for the kernel of the map from $G$ to the associated direct product to be free (one part of this result already follows from a result in S. Kim's Ph.D. thesis). The second result generalises a result of Hermiller and Sunic, again from right-angled Artin groups; we prove that for a graph $\Gamma$ with finite chromatic number, $G$ has a series in which every factor is a free product of vertex groups. The third result provides an alternative proof of a theorem due to Meier, which provides necessary and sufficient conditions on a finite graph $\Gamma$ for $G$ to be hyperbolic.