HNN extension

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

Introduced in a 1949 paper Embedding Theorems for 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' .

Let G be a group with presentation G = <S|R>, and let α : H → K be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define

The group G∗_α 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.

Since the presentation for G∗_α 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∗_α. Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of G into G∗_α. 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.

A key property of HNN-extensions is a normal form theorem known as Britton's Lemma. Let G∗_α be as above and let w be the following product in G∗_α:

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

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

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

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