Generators info

In mathematics, the HNN extension is a basic construction of combinatorial group theory.

Introduced in a 1949 paper Embeddings of Groups by Graham Higman, B. H. Neumann and Hanna Neumann, it embeds a given group G into another group G’ , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G’ .

Construction

Let <math>G</math> be a group with
presentation <math>G=\langle S\mid R\rangle</math>, and let
<math>\alpha</math> be an isomorphism between two subgroups <math>H</math> and <math>K</math> of <math>G</math>.
Let <math>t</math> be a new symbol not in <math>S</math>, and define

<math>

G*_{\alpha} = \langle S,t \mid R, tht^{-1}=\alpha(h), \forall h\in H\rangle
</math>
The group <math>G*_{\alpha}</math> is called the HNN extension of
<math>G</math> relative to <math>\alpha</math>. The original group G is called the base group for the construction, while the subgroups <math>H</math> and <math>K</math> are the associated subgroups. The new generator <math>t</math> is called the stable letter.

Key properties

Since the presentation for <math>G*_{\alpha}</math> contains all the generators and relations from the presentation for
<math>G</math>, there is a natural homomorphism, induced by the identification of generators, which takes <math>G</math> to <math>G*_{\alpha}</math>. Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of <math>G</math>
into <math>G*_{\alpha}</math>. A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.

Applications

In terms of the fundamental group in algebraic topology, the HNN extension is the construction required to understand the fundamental group of a topological space X that has been ‘glued back’ on itself by a mapping f. That is, HNN extensions stand in relation of that aspect of the fundamental group, as free products with amalgamation do with respect to the Seifert-van Kampen theorem for gluing spaces X and Y along a connected common subspace. Between the two constructions essentially any geometric gluing can be described, from the point of view of the fundamental group.

The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.

Generalizations

HNN extensions are elementary examples of fundamental groups of graphs of groups, and as such are of central importance in Bass-Serre theory.