Free groups and the tree
We saw that the Cayley graph of the free group F_S on S (with S = S⁻¹) is a regular tree: no loops because there are no relations. This is not a coincidence but a characterization. A group is free iff it acts freely (no nontrivial element fixes a vertex or inverts an edge) on a tree. From this single geometric fact tumbles one of the prettiest results in the subject.
Gluing groups: amalgams and HNN
Two constructions build new groups by gluing along subgroups. The amalgamated free product A ∗_C B takes A and B sharing a common subgroup C and freely combines them, identifying only the two copies of C. The HNN extension takes one group A with two isomorphic subgroups C₁ ≅ C₂ and adds a stable letter t that conjugates one onto the other: t⁻¹C₁t = C₂. Both have a normal form theorem letting you write elements uniquely as alternating products — the analogue of reduced words in a free group.
Examples worth memorizing:
A *_C B with C = 1: this is just the free product A * B = coproduct in Groups.
e.g. Z * Z = F_2.
PSL(2,Z) = Z/2Z * Z/3Z (amalgam over the trivial group; acts on the
Farey tree)
SL(2,Z) = Z/4Z *_{Z/2Z} Z/6Z (amalgamated over the center {+/-I})
HNN: Klein bottle group = < a, t | t a t^-1 = a^-1 >
Here A = <a> = Z, C1 = C2 = Z, the iso is a |-> a^-1.
HNN (ascending): BS(1,2) = < a, t | t a t^-1 = a^2 > (Baumslag-Solitar)
C1 = <a> = Z maps onto the index-2 subgroup C2 = <a^2>.Bass–Serre: reading the group off a tree
Here is the unifying machine. Bass–Serre theory says: a group acting on a tree (without inversions) is the same data as a graph of groups — a graph with a group attached to each vertex and edge, and injections from edge-groups into vertex-groups. The group is reconstructed as the fundamental group of that graph of groups. Amalgamated products are the case of a single edge (two vertices); HNN extensions are the case of a single loop (one vertex, one edge back to itself).