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

Lookup NU author(s): Professor Sarah Rees

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### 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 necessaryand sufficient condition on a finite graph $\Gamma$ for the kernel of the mapfrom $G$ to the associated directproduct to be free (one part of this result already follows from a resultin S. Kim's Ph.D.thesis). The second result generalisesa 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 aseries 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 finitegraph $\Gamma$ for $G$ to be hyperbolic.

Author(s): Holt DF, Rees S

Publication type: Article

Publication status: Published

Journal: Journal of Algebra

Year: 2012

Volume: 371

Pages: 94-104

Print publication date: 01/12/2012

ISSN (print): 0021-8693

ISSN (electronic): 1090-266X