Graph of groups

A graph of groups is a combinatorial structure consisting of a graph together with a group assigned to each vertex and an embedding of each edge group into its two endpoint vertex groups. Introduced by Jean-Pierre Serre in the context of Bass-Serre theory, it provides a way to decompose a group that acts on a tree into simpler pieces glued along subgroups. The fundamental theorem of Bass-Serre theory states that every group acting on a tree without edge inversions is the fundamental group of a graph of groups. This structure generalizes the free product and amalgamated product constructions, unifying them into a single geometric framework.

The graph of groups is not merely a bookkeeping device for amalgamations. It is the realization that every group with a tree-like structure is literally a space — a graph — with groups living at its points and edges. The geometry is not metaphor; it is the group's own anatomy.