Bass-Serre theory
Bass-Serre theory is a perfect dictionary between two worlds: groups acting on trees, on the one side, and groups assembled by gluing along subgroups, on the other. It says that the seemingly geometric question ‘how does my group move a tree around?’ and the seemingly algebraic question ‘how is my group built from amalgams and HNN extensions?’ are the very same question, with a precise translation each way. Where there is an action on a tree, there is a recipe for the group; where there is such a recipe, there is an action.
The organizing notion is a graph of groups: a connected graph in which each vertex and each edge carries a group, with injective edge-group maps into the two endpoint vertex groups. Its fundamental group is built by iterated amalgamated products and HNN extensions, one for each edge. The fundamental theorem states that the fundamental groups of graphs of groups are exactly the groups admitting an action on a tree (without inversions), and that the graph of groups can be recovered from the quotient graph and the vertex and edge stabilizers of the action.
Developed by Jean-Pierre Serre with Hyman Bass (Serre's 1977 book ‘Trees’), this theory makes the structure of free products, amalgams, HNN extensions, and their iterations geometrically transparent. It is the foundation for splittings of groups, accessibility results, the proof that one-relator groups and surface groups have rich subgroup structure, and the modern theory of group actions on more general trees (real trees, JSJ decompositions).
The modular group PSL(2, Z) acts on the Bass-Serre tree associated to its splitting Z/2Z * Z/3Z; the quotient is a single edge with vertex groups Z/2Z and Z/3Z and trivial edge group, recovering the free product from the geometry.
PSL(2, Z) splits as Z/2Z * Z/3Z, read off from its action on a tree.
A single edge of groups recovers the basic cases: two distinct vertices give an amalgamated free product A *_C B, while a loop (one vertex, edge from it to itself) gives an HNN extension.