nLab
HNN-extension

Idea

We have some group, $H$, with a subgroup, $A$, together with a monomorphism, $\theta :A\to H$. The HNN-extension $H{*}_A$ is obtained by adjoining an element $t$ to $H$ subject to the relations:

$$t^{-1}\mathrm{at}=\theta (a)$$t^{-1}at= \theta(a)

for all $a\in A$.

The idea is thus that the two copies of $A$ in $H$ given by $A$ itself and $\theta (A)$ become conjugate subgroups in $H{*}_A$.

Link with graphs of groups

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 $\mathrm{Gv}$ are the inclusion of $A$ into $H$ and the given monomorphism, $\theta$, then ${\Pi }_1(𝒢)(v)\cong H{*}_A$.

References

• J.-P. Serre, 1977, Arbres, amalgames, ${\mathrm{SL}}_2$, volume 46 of Astérisque , Société mathématique de France.

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