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

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Abstract

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. Kim's 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. © 2012 Elsevier Inc.

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Holt, D. F., & Rees, S. (2012). Generalising some results about right-angled Artin groups to graph products of groups. Journal of Algebra, 371, 94–104. https://doi.org/10.1016/j.jalgebra.2012.07.049

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