HNN extension
The HNN extension group G for a group A, is constructed from a pair of isomorphic subgroups
Bϕ≅C in A, according to formula
G=A*⟨t|-⟩N |
where
⟨t|-⟩ is a cyclic free group, * is the free product
and N is the normal closure
of {tbt-1ϕ(b)-1:b∈B}.
As an example take a surface bundle F⊂E→S1, hence the homotopy long exact sequence of this bundle implies that the fundamental group
π1(E) is given by
π1(E)=⟨x1,…,xk,t|Π=1,txit-1=ϕ(xi)⟩ |
where k is the genus of the surface and the relation Π is [x1,x2][x3,x4]⋯[xk-1,xk] for an orientable surface or x21x22⋯x2k is for a non-orientable one. ϕ 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 |