nLab graph of groups

Idea

Given a group action on a graph $Y$ (with involution on the edges), one gets a quotient graph $X=Y/G$ together with the family of the stabilisers on the vertices and on the edges. Abstracting this gives a graph of groups.

Definition

A graph of groups, $\Gamma(G,X)$, consists of

• a connected graph, $X$;

• a function, $G$, which, for every vertex, $v\in V(X)$ assigns a group, $G_v$, and for each edge, $e\in E(X)$ assigns a group, $G_e$ such that $G_e= G_\overline{e}$

• For each edge, $e\in E(X)$, there exists a monomorphism, $\sigma::G_e\to G_{i(e)}$, where $i(e)$ is the initial vertex of the edge, $e$.

Examples

In the usual form of Bass-Serre Theory a graph of groups is derived from the action of a group on a tree.

Related entries.

• complex of groups - The theory of graphs of groups is the 1-dimensional precursor of that of complexes of groups.

References

• J.-P. Serre, 1977, Arbres, amalgames, SL2, volume 46 of Astérisque, Société mathématique de France.

• J.-P. Serre, 2003, Trees, Springer Monographs in Mathematics, Springer- Verlag Berlin

• H. Bass, Covering theory for graphs of groups, Journal of Pure and Applied Algebra, 89, (1993), 3–47.