Graph of groups

In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabiliser subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.

A graph of groups over a graph Y is an assignment to each vertex x of Y of a group G_x and to each edge y of Y of a group G_y as well as monomorphisms φ_y,0 and φ_y,1 mapping G_y into the groups assigned to the vertices at its ends.

Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups G_x and elements y for each edge of Y with the following relations:

This definition is independent of the choice of T.

The benefit in defining the fundamental groupoid of a graph of groups, as shown by Higgins (1976), is that it is defined independently of base point or tree. Also there is proved there a nice normal form for the elements of the fundamental groupoid. This includes normal form theorems for a free product with amalgamation and for an HNN extension (Bass 1993).

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