HNN extension


In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard 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 in G' .

Construction

Let G be a group with presentation, and let be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define
The group is called the HNN extension of ''G relative to α. The original group G is called the base group for the construction, while the subgroups H'' and K are the associated subgroups. The new generator t is called the stable letter.

Key properties

Since the presentation for contains all the generators and relations from the presentation for G, there is a natural homomorphism, induced by the identification of generators, which takes G to. Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into. 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.

Britton's Lemma

A key property of HNN-extensions is a normal form theorem known as Britton's Lemma. Let be as above and let w be the following product in :
Then Britton's Lemma can be stated as follows:
Britton's Lemma. If w = 1 in G∗α then
  • either and g0 = 1 in G
  • or and for some i ∈ one of the following holds:
  1. εi = 1, εi+1 = −1, gi ∈ H,
  2. εi = −1, εi+1 = 1, gi ∈ K.

In contrapositive terms, Britton's Lemma takes the following form:
Britton's Lemma. If w is such that
  • either and g0 ≠ 1 ∈ G,
  • or and the product w does not contain substrings of the form tht−1, where h ∈ H and of the form t−1kt where k ∈ K,
then in.

Consequences of Britton's Lemma

Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:
  • The natural homomorphism from G to is injective, so that we can think of as containing G as a subgroup.
  • Every element of finite order in is conjugate to an element of G.
  • Every finite subgroup of is conjugate to a finite subgroup of G.
  • If contains an element such that is contained in neither nor for any integer, then contains a subgroup isomorphic to a free group of rank two.

    Applications and generalizations

Applied to algebraic topology, the HNN extension constructs the fundamental group of a topological space X that has been 'glued back' on itself by a mapping f : X → X. Thus, HNN extensions describe the fundamental group of a self-glued space in the same way that free products with amalgamation do for two spaces X and Y glued along a connected common subspace, as in the Seifert-van Kampen theorem. The HNN extension is a natural analogue of the amalgamated free product, and comes up in determining the fundamental group of a union when the intersection is not connected. These two constructions allow the description of the fundamental group of any reasonable geometric gluing. This is generalized into the Bass–Serre theory of groups acting on trees, constructing fundamental groups of graphs of groups.
HNN-extensions play a key role in Higman's proof of the Higman embedding theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group. Most modern proofs of the Novikov–Boone theorem about the existence of a finitely presented group with algorithmically undecidable word problem also substantially use HNN-extensions.
The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.