# HNN extension

The HNN extension group $G$ for a group $A$, is constructed from a pair of isomorphic subgroups $B\lx@stackrel{{\scriptstyle\phi}}{{\cong}}C$ in $A$, according to formula

 $G=\frac{A*\langle t|-\rangle}{N}$

where $\langle t|-\rangle$ is a cyclic free group, $*$ is the free product and $N$ is the normal closure of $\{tbt^{-1}\phi(b)^{-1}\colon b\in B\}$.

As an example take a surface bundle $F\subset E\to S^{1}$, hence the homotopy long exact sequence of this bundle implies that the fundamental group $\pi_{1}(E)$ is given by

 $\pi_{1}(E)=\langle x_{1},...,x_{k},t|\Pi=1,tx_{i}t^{-1}=\phi(x_{i})\rangle$

where $k$ is the genus of the surface and the relation $\Pi$ is $[x_{1},x_{2}][x_{3},x_{4}]\cdots[x_{k-1},x_{k}]$ for an orientable surface or $x_{1}^{2}x_{2}^{2}\cdots x_{k}^{2}$ is for a non-orientable one. $\phi$ is an isomorphism induced by a self homeomorphism of $F$.

Title HNN extension HNNExtension 2013-03-22 16:04:03 2013-03-22 16:04:03 juanman (12619) juanman (12619) 8 juanman (12619) Definition msc 20E06 GroupExtension