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:

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?, $\mathcal{G}$, 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 $\mathrm{Gv}$ 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, ${\mathrm{SL}}_{2}$, volume 46 of Astérisque , Société mathématique de France.

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

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