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 (through a given isomorphism) 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 G*_{\alpha} 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 G_{\alpha} 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 G{\alpha}. Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into G*{\alpha}. 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 G_{\alpha} be as above and let w be the following product in G{\alpha}: Then Britton's Lemma can be stated as follows: Britton's Lemma. If w = 1 in G∗α then In contrapositive terms, Britton's Lemma takes the following form: Britton's Lemma (alternate form). If w is such that then w\ne 1 in G*{\alpha}.

Consequences of Britton's Lemma

Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:

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 (see e.g. Surface bundle over the circle). 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.