Normal form theorem for an HNN extension

Fix right coset transversals $R_{+},R_{-}$ for $A,B$, containing $1$. In the convention $ta{t}^{-1}=\varphi (a)$, every element has a unique normal form $g_0{t}^{{ϵ}_1}{r}_1\cdots t^{{ϵ}_k}{r}_k$, with $r_i\in R_{{ϵ}_i}$ and $r_i\ne 1$ whenever ${ϵ}_{i+1}=-{ϵ}_i$. A permutation action on the set of these formal sequences proves uniqueness and embeds the base group. Transporting subgroup coefficients across stable letters converts general words to these normal forms.

A reduced sequence is a word $g_0{t}^{{ϵ}_1}{g}_1\cdots t^{{ϵ}_k}{g}_k$ with no pinch in an HNN extension. If $k>0$, Britton's lemma makes it nontrivial and prevents it from representing a base-group element. Reduced sequences need not be unique; uniqueness requires the fixed coset representatives in the normal form theorem for an HNN extension.

A cyclically reduced sequence has no pinch in an HNN extension either internally or across its cyclic junction. If its stable-letter length is positive, every positive power remains reduced, so Britton's lemma gives infinite order. Repeated cyclic conjugation and pinch removal shows that any finite-order element of an HNN extension is conjugate into its base group.

In $⟨G,t\mid ta{t}^{-1}=\varphi (a),a\in A⟩$, a pinch is a subword $ta{t}^{-1}$ with $a\in A$, or $t^{-1}bt$ with $b\in B=\varphi (A)$. It can be replaced by a base-group element, removing two stable letters. Britton's lemma asserts that every identity word with stable letters must have a pinch after base-group multiplication is performed.