nLab
graph of groups

Contents

Contents

Idea

Given a group action on a graph YY (with involution on the edges), one gets a quotient graph X=Y/GX=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, Γ(G,X)\Gamma(G,X), consists of

  • a connected graph, XX;

  • a function, GG, which, for every vertex, vV(X)v\in V(X) assigns a group, G vG_v, and for each edge, eE(X)e\in E(X) assigns a group, G eG_e such that G e=G e¯G_e= G_\overline{e}

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

Examples

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

  • 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.

Last revised on February 19, 2020 at 06:37:04. See the history of this page for a list of all contributions to it.