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

Derek F. Holt and Sarah Rees

Keywords

graph product, right angled Artin group, hyperbolic group

Status

submitted for publication

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 \u{S}uni\'{c}, 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. \

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