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

Abstract

We prove three results about the graph product $G=\G(Γ;G_v, v \in V(Γ))$ of groups $G_v$ over a graph $Γ$. 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 $Γ$ 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 $Γ$ 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 $Γ$ for $G$ to be hyperbolic.