Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an automorphism $F_{f}$ of $\hat{F}_{2}$, and which furthermore satisfy the following three relations:
$f\left(a_{2}^{2}, a_{1}^{2}\right) f\left(a_{1}^{2}, a_{2}^{2}\right)=1$ in $\hat{\Gamma}_{1}^{1}$, where $\alpha_{1}$ and $\alpha_{2}$ are as in figure $1(\mathrm{a})$;
$f\left(b_{3}, b_{1}\right) f\left(b_{2}, b_{3}\right) f\left(b_{1}, b_{2}\right)=1$ in $\hat{\Gamma}_{0}^{4}$, where $\beta_{1}, \beta_{2}$ and $\beta_{3}$ are as in figure $1(\mathrm{~b})$;
$f\left(b_{3}, b_{4}\right) f\left(b_{5}, b_{1}\right) f\left(b_{2}, b_{3}\right) f\left(b_{4}, b_{5}\right) f\left(b_{1}, b_{2}\right)=1$ in $\hat{\Gamma}_{0}^{5}$, where the $\beta_{i}$ are as in figure $1(\mathrm{c})$.
In this definition, $f(a, b)$ denotes the image of $f$ under a homomorphism of $\hat{F}_{2}$ into some profinite group $G$ sending $x \mapsto a$ and $y \mapsto b$. The set $\widehat{G T}^{1}$ is made into a group by defining the multiplication law $f \cdot g$ to be given by composition of the automorphisms. In other words, if $F_{f}$ and $F_{g}$ denote the automorphisms of $\hat{F}_{2}$ associated to $f$ and $g \in \widehat{G T}^{1}$ then the automorphism $F_{g f}$ is defined to be $F_{g} \circ F_{f}$, so that we have $g \cdot f=g F_{g}(f)$
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