We have some group, , with a subgroup, , together with a monomorphism, . The HNN-extension is obtained by adjoining an element to subject to the relations:
for all .
The idea is thus that the two copies of in given by itself and become conjugate subgroups in .
Examine the fundamental group? of the graph of groups?, , with underlying graph the graph with one vertex, and one edge, and nothing else. Take the vertex group, , to be , the edge group, , to be , and the two morphisms from to are the inclusion of into and the given monomorphism, , then .
J.-P. Serre, 1977, Arbres, amalgames, , volume 46 of Astérisque , Société mathématique de France.
J.-P. Serre, 2003, Trees , Springer Monographs in Mathematics, Springer-Verlag Berlin.
150, 157, 161