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

t 1at=θ(a)t^{-1}at= \theta(a)

for all aAa\in A.

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

Examine the fundamental group? of the graph of groups, 𝒢\mathcal{G}, with underlying graph the graph with one vertex, vv and one edge, ee and nothing else. Take the vertex group, G vG_v, to be HH, the edge group, G eG_e, to be AA, and the two morphisms from G eG_e to GvGv are the inclusion of AA into HH and the given monomorphism, θ\theta, then Π 1(𝒢)(v)H* A\Pi_1(\mathcal{G})(v) \cong H*_A.


  • J.-P. Serre, 1977, Arbres, amalgames, SL 2SL_2, volume 46 of Astérisque , Société mathématique de France.

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

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