HNN extension

In mathematics, the HNN extension is an important construction of combinatorial group theory.

Introduced in a 1949 paper Embedding Theorems for Groups^[1] 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 ${\displaystyle G=\langle S\mid R\rangle }$, and let ${\displaystyle \alpha \colon H\to K}$ be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define

${\displaystyle G*_{\alpha }=\left\langle S,t\mid R,tht^{-1}=\alpha (h),\forall h\in H\right\rangle .}$

The group ${\displaystyle G*_{\alpha }}$ 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 ${\displaystyle G*_{\alpha }}$ 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 ${\displaystyle G*_{\alpha }}$. Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into ${\displaystyle G*_{\alpha }}$. 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.^[2] Let ${\displaystyle G*_{\alpha }}$ be as above and let w be the following product in ${\displaystyle G*_{\alpha }}$:

${\displaystyle w=g_{0}t^{\varepsilon _{1}}g_{1}t^{\varepsilon _{2}}\cdots g_{n-1}t^{\varepsilon _{n}}g_{n},\qquad g_{i}\in G,\varepsilon _{i}=\pm 1.}$

Then Britton's Lemma can be stated as follows:

Britton's Lemma. If w = 1 in G∗_α then

• either ${\displaystyle n=0}$ and g₀ = 1 in G
• or ${\displaystyle n>0}$ and for some i ∈ {1, ..., n−1} one of the following holds:

1. ε_i = 1, ε_i+1 = −1, g_i ∈ H,
2. ε_i = −1, ε_i+1 = 1, g_i ∈ K.

In contrapositive terms, Britton's Lemma takes the following form:

Britton's Lemma (alternate form). If w is such that

• either ${\displaystyle n=0}$ and g₀ ≠ 1 ∈ G,
• or ${\displaystyle n>0}$ 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 ${\displaystyle w\neq 1}$ in ${\displaystyle G*_{\alpha }}$.

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 ${\displaystyle G*_{\alpha }}$ is injective, so that we can think of ${\displaystyle G*_{\alpha }}$ as containing G as a subgroup.
• Every element of finite order in ${\displaystyle G*_{\alpha }}$ is conjugate to an element of G.
• Every finite subgroup of ${\displaystyle G*_{\alpha }}$ is conjugate to a finite subgroup of G.
• If ${\displaystyle H\neq G}$ and ${\displaystyle K\neq G}$ then ${\displaystyle G*_{\alpha }}$ contains a subgroup isomorphic to a free group of rank two.

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 (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.

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.

Both HNN-extensions and amalgamated free products are basic building blocks in the Bass–Serre theory of groups acting on trees.^[3]

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.

References[]