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 G = ⟨ S ∣ R ⟩ {\displaystyle G=\langle S\mid R\rangle } , and let α : H → K {\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
G ∗ α = ⟨ S , t ∣ R , t h t − 1 = α ( h ) , ∀ h ∈ H ⟩ . {\displaystyle G*_{\alpha }=\left\langle S,t\mid R,tht^{-1}=\alpha (h),\forall h\in H\right\rangle .}
The group G ∗ α {\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 G ∗ α {\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 G ∗ α {\displaystyle G*_{\alpha }} . Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into G ∗ α {\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. Let G ∗ α {\displaystyle G*_{\alpha }} be as above and let w be the following product in G ∗ α {\displaystyle G*_{\alpha }} :
w = g 0 t ε 1 g 1 t ε 2 ⋯ g n − 1 t ε n g n , g i ∈ G , ε i = ± 1. {\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 n = 0 {\displaystyle n=0} and g0 = 1 in G or n > 0 {\displaystyle n>0} and for some i ∈ {1, ..., n−1} one of the following holds: εi = 1, εi+1 = −1, gi ∈ H, εi = −1, εi+1 = 1, gi ∈ K.
In contrapositive terms, Britton's Lemma takes the following form:
Britton's Lemma (alternate form). If w is such that either n = 0 {\displaystyle n=0} and g0 ≠ 1 ∈ G, or n > 0 {\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 w ≠ 1 {\displaystyle w\neq 1} in G ∗ α {\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:
… excerpt ends here. Continue reading the full article.
