# HNN extension

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

$$G=\frac{A*\u27e8t|-\u27e9}{N}$$ |

where
$\u27e8t|-\u27e9$ is a cyclic free group^{}, $*$ is the free product^{} and $N$ is the normal closure^{} of $\{tb{t}^{-1}\varphi {(b)}^{-1}:b\in B\}$.

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

$${\pi}_{1}(E)=\u27e8{x}_{1},\mathrm{\dots},{x}_{k},t|\mathrm{\Pi}=1,t{x}_{i}{t}^{-1}=\varphi ({x}_{i})\u27e9$$ |

where $k$ is the genus of the surface and the relation^{} $\mathrm{\Pi}$ is $[{x}_{1},{x}_{2}][{x}_{3},{x}_{4}]\mathrm{\cdots}[{x}_{k-1},{x}_{k}]$ for an orientable surface or ${x}_{1}^{2}{x}_{2}^{2}\mathrm{\cdots}{x}_{k}^{2}$ is for a non-orientable one. $\varphi $ 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 |