# nLab HNN-extension

## Idea

We have some group, $H$, with a subgroup, $A$, together with a monomorphism, $\theta : A\to H$. The HNN-extension $H\ast_A$ 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 $H\ast_A$.

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 18:57:10 by Tim Porter (95.147.236.147)