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

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

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 $H\ast_A$.

Link with graphs of groups

Examine the fundamental group? of the graph of groups, $\mathcal{G}$, 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, $\theta$, then $\Pi_1(\mathcal{G})(v) \cong H*_A$.

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 at 18:57:10.
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