nLab graph product

Idea

Given two groups, two typical ways of combining them into a new group is by taking either their direct product or their free product. Given a finite family of groups, you could repeatedly take binary direct products to get the direct product of the whole family, or repeatedly take binary free products to get the free product of the whole family. One could if they wish take a mix of direct products and free products. Given a graph and a group assigned to each vertex, the graph product of this family of groups is a group which looks like a direct product if you look at two groups connected by an edge, and a free product otherwise.

Definition

Let $\Gamma$ be a simple graph with vertex set $V$ and edge set $E \subseteq \binom{V}{2}$, and $\{G_v\}_{v \in V}$ a family of groups indexed by the vertex set. The graph product group $\Gamma G$ is given by first taking the free product of the groups, and then modding out by the normal subgroup generated by the relation which imposes commutativity for those elements which correspond to groups which are connected by an edge in the graph, i.e.

$$\Gamma G = \coprod G_v / R_\Gamma$$

where $R_\Gamma$ is the normal subgroup generated by the relation $g_v g_u \sim g_u g_v$ for $\{u,v\} \in E$, $g_u \in G_u$, and $g_v \in G_v$.

Examples

• If the graph has no edges, one recovers free products. If the graph is complete, one recovers direct products.

• Right-angled Artin groups

References

The notion of graph products of groups was introduced by Elisabeth Green.

• Elisabeth Green?, Graph products of groups, PhD thesis, University of Leeds, (1990). pdf

António Veloso da Costa generalized the concept to monoids.

• António Veloso da Costa?, Graph products of monoids, Semigroup Forum, Volume 63, Issue 2, pp 247–277 (2001). web

Right-angled Artin groups

• Ruth Charney?, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141–158. (arxiv:0610668)