HNN extension

The HNN extension group G for a group A, is constructed from a pair of isomorphicPlanetmathPlanetmathPlanetmath subgroupsMathworldPlanetmathPlanetmath BϕC in A, according to formula


where t|- is a cyclic free groupMathworldPlanetmath, * is the free productMathworldPlanetmath and N is the normal closurePlanetmathPlanetmath of {tbt-1ϕ(b)-1:bB}.

As an example take a surface bundle FES1, hence the homotopyMathworldPlanetmathPlanetmath long exact sequence of this bundle implies that the fundamental groupMathworldPlanetmathPlanetmath π1(E) is given by


where k is the genus of the surface and the relationPlanetmathPlanetmathPlanetmath Π is [x1,x2][x3,x4][xk-1,xk] for an orientable surface or x12x22xk2 is for a non-orientable one. ϕ is an isomorphismMathworldPlanetmathPlanetmathPlanetmath 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