minimal and maximal subscripts that appear when we rewrite $r$ as a word $r^*$ in $B^*\cup C^*$. With this notation $G^* = \langle B^*,C^* \ | \ r^* =1 \rangle$ and the two edge groups are $L= F(B^*,C^*_-), U=F(B^*,C^*_+)$, where $C^*_- = \{c_{\mu},  \ldots, c_{\nu-1}\}$ and $C^*_+ = \{c_{\mu+1},  \ldots, c_{\nu}\}$. We note that by requiring that $r$ begins with $c^{\pm 1}$ we have ensured that $\mu \le 0 \leq \nu$. (We do not exclude the possibility that $\mu=0=\nu$ in which case $C^*_+$ and $C^*_-$ are both empty but we will not usually make explicit reference to this since the argument is either unchanged or even simplified.) Given any $z \in U$ we write $\overleftarrow z$ for the word obtained by reducing subscripts by one and similarly for any $w \in L$, we write $\overrightarrow w$ when we increase the subscripts by one.

We can transform any equality $gh(A,B)g^{-1}=k(B,C)$ into one expressed in the generators of $G$ as HNN-extension. We write $g=g_0a^{\varepsilon_1}g_1\ldots a^{\varepsilon_m}g_m$ in reduced form, where $\varepsilon_i = \pm 1$. Since $k$ omits $a$, $h$ has zero exponent sum in $a$ and thus both $h$ and $k$ lie in the base group $G^*$ – $h \in F(B^*)$ and $k 