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
Canonical name	HNNExtension
Date of creation	2013-03-22 16:04:03
Last modified on	2013-03-22 16:04:03
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	8
Author	juanman (12619)
Entry type	Definition
Classification	msc 20E06
Related topic	GroupExtension