# graph product of groups

Let $\Gamma$ be a finite undirected graph and let $\{G_{v}\colon v\in V(\Gamma)\}$ be a collection  of groups associated with the vertices of $\Gamma$. Then the graph product  of the groups $G_{v}$ is the group $G=F/R$, where $F$ is the free product of the $G_{v}$ and $R$ is generated by the relations   that elements of $G_{u}$ commute with elements of $G_{v}$ whenever $u$ and $v$ are adjacent in $\Gamma$.

The free product and the direct product   are the extreme examples of the graph product. To obtain the free product, let $\Gamma$ be an anticlique, and to obtain the direct product, let $\Gamma$ be a clique.

## References

• 1 E.R. Green, Graph products of groups, Doctoral thesis, The University of Leeds, 1990.
• 2 S. Hermiller and J. Meier, Algorithms and geometry for graph products of groups, Journal of Algebra 117 (1995), 230–257.
• 3 M. Lohrey and G. Sénizergues, When is a graph product of groups virtually-free?, to appear in Communications in Algebra. 2006 preprint available online at http://inf.informatik.uni-stuttgart.de/fmi/ti/personen/Lohrey/05-Graphprod.pdf.
• 4 R.Brown, M. Bullejos, and T. Porter,‘Crossed complexes, free crossed resolutions and graph products of groups’, Proceedings Workshop Korea 2000, J. Mennicke, Moo Ha Woo (eds.) Recent Advances in Group Theory, Heldermann Verlag Research and Exposition in Mathematics 27 (2002) 8–23. arXiv:math/0101220
Title graph product of groups GraphProductOfGroups 2013-03-22 16:10:36 2013-03-22 16:10:36 mps (409) mps (409) 8 mps (409) Definition msc 20F65 graph product of groups graph product