Bass–Serre theory

Summary

Bass–Serre theory describes the groups that act on trees by building them from simpler groups. Its central object is a graph of groups: a graph with a group at each vertex and each edge. The structure theorem says that a group acting on a tree without inversions is the fundamental group of the quotient graph of groups. The vertex and edge groups of this graph of groups are stabilisers. Conversely, the fundamental group of a graph of groups acts on a tree, the Bass–Serre tree. It unifies free products, amalgamated free products and HNN extensions. It gives short proofs of subgroup theorems such as the Nielsen–Schreier theorem and the Kurosh subgroup theorem.