Understanding HNN extensions: intuition, examples, exercises.

Question

What is an HNN extension? What would be some elementary, intuitive examples of them and what exercises involving them would you suggest?

The Wikipedia definition is easiest to get to, since neither indexes of Magnus et al. nor Johnson's "Presentation$\color{red}{s}$ of Groups (Old Version)" indicate where they are.

Here it is for convenience:

Let $G$ be a group with presentation $G=\langle S\mid R\rangle$, and let $\alpha: H\to K$ be an isomorphism between two subgroups of $G$. Let $t$ be a symbol not in $S$, and define

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

The group $G\ast_\alpha$ is called the HNN extension of $G$ relative to $\alpha$. 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.

The definition in Lyndon & Schupp's "Combinatorial Group Theory", despite the concept being mentioned a few times prior to it, is on page 179. It's very similar:

Let $G$ be a group, and let $A$ and $B$ be subgroups of $G$ with $\phi: A\to B$ an isomorphism. The HNN extension of $G$ relative to $A$ and $B$ and $\phi$ is the group

$$G^\ast=\langle G, t\mid t^{-1}at=\phi(a), a\in A\rangle.$$

The group $G$ is called the base of $G^\ast$, $t$ is called the stable letter, and $A$ and $B$ are called the associated subgroups.

The definition in Baumslag's "Topics in Combinatorial Group Theory", page 66, reads

Definition 2: Let $$B=\langle X\mid R\rangle$$ be a group given by a presentation and suppose $U$ and $V$ are subgroups of $B$ equipped with an isomorphism

$$\tau: U\stackrel{\sim}{\longrightarrow} V.$$

Then we term

$$E=\langle X, t\mid R\cup\{ tut^{-1}=u\tau\}\rangle\quad(\text{where }t\notin X)$$

an HNN extension of $B$ with stable letter $t$, associated subgroups $U$ and $V$ and associating isomorphism $\tau$.

These are all well & good. I can see how they are equivalent quite readily.

The definition in Stillwell's "Classic Topology and Combinatorial Group Theory (Second Edition)" is on page 286 and is much different. I shan't copy it down, for it's quite lengthy, but, again, I can sort of see what it's getting at. There are three exercises on HNN extensions starting on that page too, now that I've looked at the dreaded topology book, so I'll give them a go now.

If you have anything more to add to help me and others understand this important concept, please do so as a comment or even an answer.

I still don't get the intuition behind'm and I know not of any tangible examples yet.

Please help :)

Answer 1

Here's a few bits and bobs.

1) HNN extensions generalize semidirect products with $\mathbb Z$: Take $A$ to be $G$ and $\phi$ to be an automorphism of $G$. Then, $G\ast_\phi = G\rtimes_\phi \mathbb Z$

2) The inclusion of $G$ into any HNN extension $G\ast_\alpha$ is injective, so we can think of the HNN extension as "adding something" on top of our group.

3) Moreover, HNN extensions make more things conjugate in $G\ast_\alpha$ than were in $G$. This is an interesting property to play with!

4) Some of my favourite examples (some of which are downright strange):

• The Baumslag-Solitar groups $BS(n,m) = \langle a, t \mid ta^nt^{-1} = a^m\rangle$ (which include $\mathbb Z\oplus \mathbb Z$ as $BS(1,1)$ and the Klein bottle group as $BS(1,-1)$).

• Some automorphisms of free groups can be rewritten as HNN extensions over $\mathbb Z \oplus \mathbb Z$, for example the automorphism $a \mapsto a$ and $b \mapsto ba$ yields the HNN extension (with stable letter $b$!) $F_2\rtimes\mathbb Z = \langle a,b, t \mid tat^{-1} = a, tbt^{-1} = ba\rangle = \langle a, t, b \mid [a,t], btb^{-1} = at\rangle = (\mathbb Z \oplus \mathbb Z)\ast$

• Thompson's group $F$ is an HNN extension of itself, despite admitting a finite presentation!