Graph of groups

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In geometric group theory, a graph of groups attaches a group to each vertex and each edge of a graph, together with maps from the edge groups into the vertex groups at the ends of the edges.

There is a single fundamental group associated to every finite connected graph of groups. This group acts on a tree in a way that the original graph can be recovered from the quotient tree and the stabilizer subgroups. This is the core idea of Bass–Serre theory.

More concretely, a graph of groups over a graph Y assigns a group to each vertex x (Gx) and to each edge y (Gy), along with monomorphisms from Gy into the vertex groups at the ends of y. If T is a spanning tree of Y, one defines the fundamental group Γ as the group generated by the vertex groups and by elements corresponding to edges, with relations expressing how the edge groups sit inside the vertex groups. The construction does not depend on the choice of T. The theory also has a groupoid version: the fundamental groupoid is defined independently of any base point or tree and comes with a convenient normal form for its elements, including for familiar constructions like free products with amalgamation and HNN extensions.

When you fix a spanning tree T and look at Γ, each vertex and edge group embeds into Γ. One can build a graph whose vertices are the cosets Γ/Gx and Γ/Gy and whose edges reflect the relations given by the edge monomorphisms. This graph is a tree, called the universal covering tree, on which Γ acts. The original graph of groups appears as the quotient by this action, and the stabilizers in the action recover the original vertex and edge groups.

A natural generalization is a 2-dimensional complex of groups. These are modeled on orbifolds arising from cocompact, properly discontinuous actions of discrete groups on 2-dimensional CAT(0) simplicial complexes. The quotient complex has finite stabilizer groups at vertices, edges, and 2-simplices, with monomorphisms for every inclusion. A complex of groups is developable if it comes from a CAT(0) complex. Developability is a non-positive curvature condition that can be checked locally by looking at the links of vertices, where every circuit must have length at least six.

These ideas originated in the study of 2-dimensional Bruhat–Tits buildings and have been broadened by developments in the field, with influence from Gromov’s ideas.