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

### ncl

#### Keywords

graph products, hyperbolic groups

#### Status

published in Journal of Algebra

#### 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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