HNN extension
Suppose a group contains two subgroups that are abstractly isomorphic but sit in different places — like the left and right halves of a structure that mirror each other. An HNN extension adds a new element, a ‘stable letter,’ whose only job is to conjugate one subgroup onto the other, physically realizing that isomorphism as an inner automorphism in a larger group. It is the self-gluing counterpart of the amalgamated product, which glued two different groups.
Let G be a group with two subgroups H and K and an isomorphism φ : H to K. The HNN extension is G*_φ = ⟨ G, t | t^(-1) h t = φ(h) for all h in H ⟩, where t is a new generator (the stable letter). Britton's lemma gives the normal form: a reduced word g_0 t^(e_1) g_1 ... t^(e_n) g_n with no ‘pinch’ (no subword t^(-1) g t with g in H, nor t g t^(-1) with g in K) represents a nontrivial element. In particular G embeds in G*_φ.
Higman, B. H. Neumann and H. Neumann introduced these in 1949 to prove that every countable group embeds in a 2-generator group, and HNN extensions are central to the unsolvability of the word problem (Novikov-Boone) precisely because they can build wild groups from tame ones. Together with amalgamated products they are the two operations from which Bass-Serre theory reconstructs all groups acting on trees.
The Baumslag-Solitar group BS(m, n) = ⟨ a, t | t^(-1) a^m t = a^n ⟩ is the HNN extension of Z with H = mZ, K = nZ, and φ multiplying by n/m. BS(1, 2) is solvable but not residually finite-friendly in the naive sense, illustrating how HNN extensions manufacture exotic behavior.
Baumslag-Solitar groups BS(m, n) are the canonical examples of HNN extensions.
Topologically, an HNN extension is the fundamental group of a space built by attaching a cylinder S^1 × [0,1] to a space for G, gluing the two ends along the loops representing H and K.