An HNN extension with its stable letter

Let $A$ and $C$ be groups, and let

$\alpha ,\beta :C\to A$

be injective group homomorphisms (Monoid homomorphism and group homomorphism). Let $F(\{t\})$ be the free group on one generator and form the free product $A\ast F(\{t\})$ (The free product of an arbitrary family of groups, Free group on a set of generators). The HNN extension of $A$ with associated edge maps $\alpha ,\beta$ is

$\frac{A\ast F(\{t\})}{⟨\text{\hspace{0.17em}⁣}⟨t\alpha (c)t^{-1}\beta (c{)}^{-1}:c\in C⟩\text{\hspace{0.17em}⁣}⟩}\mathrm{.}$

Equivalently, in relative-presentation notation, it is

$⟨A,t|t\alpha (c)t^{-1}=\beta (c)\text{ for every }c\in C⟩\mathrm{.}$

Thus the conjugacy relations are

$t\alpha (c)t^{-1}=\beta (c)(c\in C)\mathrm{.}$

If $A=⟨X\mid R⟩$ is presented as in Group presentation by generators and relations, choose words $u_c,v_c\in F(X)$ representing $\alpha (c),\beta (c)$ for each $c\in C$. The corresponding ordinary group presentation is

$⟨X,t|R,t{u}_c{t}^{-1}{v}_c^{-1}\text{ for every }c\in C⟩\mathrm{.}$

Different choices of $u_c,v_c$ differ by consequences of $R$, so they give the same quotient. The relative notation avoids making any word-lift choice.

The new generator $t$ is the stable letter.