---
tag: TACX13XZST
kind: definition
name: HNN extension
msc: ["20E06", "20F05"]
defines: YEYERRY37K
uses: [G5S1M9, 0X1JXT47GA, 08MG7H8JW5, 0JGEQXQ4AC, TM7QS7P7X9, 0CB0DX5BVV]
sources:
  - url: https://en.wikipedia.org/wiki/HNN_extension
---

## Definition

Let $G$ be a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] $\langle S \mid R \rangle$.
Let $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$, and let $\alpha \colon H \to K$ be a group [[isomorphism|08MG7H8JW5]].
Let $t$ be a new symbol not in $S$.
The *HNN extension* of $G$ relative to $\alpha$ is the group
$$G *_{\alpha} = \langle S, t \mid R,\ t h t^{-1} = \alpha(h) \text{ for all } h \in H \rangle,$$
where each $h \in H$ and $\alpha(h) \in K$ is written as a word in $S$.
Equivalently, $G *_{\alpha}$ is the quotient of the [[free product|TM7QS7P7X9]] $G * \langle t \rangle$, with $\langle t \rangle$ infinite cyclic, by the [[normal subgroup|0CB0DX5BVV]] generated by the elements $t h t^{-1} \alpha(h)^{-1}$ for $h \in H$.
The element $t$ is the *stable letter*, and $H$ and $K$ are the *associated subgroups*.

## Remarks

The natural map from $G$ to $G *_{\alpha}$ is injective, so $G$ is a subgroup of $G *_{\alpha}$; this is a theorem of Higman, Neumann and Neumann.
Britton's lemma describes when a word in $S$ and $t$ represents the identity.

Example: let $G = \langle a \mid \ \rangle \cong \Z$, $H = \langle a^m \rangle$ and $K = \langle a^n \rangle$ with $m, n$ nonzero integers, and $\alpha(a^m) = a^n$.
The map $\alpha$ is an isomorphism, since both subgroups are infinite cyclic and $\alpha$ sends a generator to a generator.
The HNN extension is $\langle a, t \mid t a^m t^{-1} = a^n \rangle$, the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(m, n)$.
For $m = n = 1$ it is $\langle a, t \mid t a t^{-1} = a \rangle \cong \Z^2$.

Non-example: the rule $t h t^{-1} = \varphi(h)$ for a homomorphism $\varphi \colon H \to K$ that is not injective is not an HNN extension.
If $\varphi(h) = 1$ for some $h \ne 1$, then $h = t^{-1} \varphi(h) t = 1$ in the quotient, so $G$ does not embed.