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

We prove three results about the graph product G=G(Γ;Gv,v∈V(Γ)) of groups Gv 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. Kimʼs PhD thesis). The second result generalises a result of Hermiller and Šunić, 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.