nLab
HNN-extension

Idea

We have some group, H, with a subgroup, A, together with a monomorphism, θ:AH. The HNN-extension H* A is obtained by adjoining an element t to H subject to the relations:

t 1at=θ(a)t^{-1}at= \theta(a)

for all aA.

The idea is thus that the two copies of A in H given by A itself and θ(A) become conjugate subgroups in H* A.

Examine the fundamental group? of the graph of groups?, 𝒢, with underlying graph the graph with one vertex, v and one edge, e and nothing else. Take the vertex group, G v, to be H, the edge group, G e, to be A, and the two morphisms from G e to Gv are the inclusion of A into H and the given monomorphism, θ, then Π 1(𝒢)(v)H* A.

References

  • J.-P. Serre, 1977, Arbres, amalgames, SL 2, volume 46 of Astérisque , Société mathématique de France.

  • J.-P. Serre, 2003, Trees , Springer Monographs in Mathematics, Springer-Verlag Berlin.
    150, 157, 161

Created on April 3, 2012 18:57:10 by Tim Porter (95.147.236.147)