Proposition 2

If N is a distinguished subgroup of $G_a$, the set of elements $\mathbf{z}\in \Omega (\widetilde{\Gamma})$ such that $c(\mathbf{z})$ is contained in N is a distinguished subset of $\Omega (\widetilde{\Gamma})$.

Let Z denote the set of elements $\mathbf{z}\in \Omega (\widetilde{\Gamma})$ such that $c(\mathbf{z})\subset N$.

Let $z\in$ Fl($\widetilde{\Gamma}$). We have $c(z, z) =\{e_a\}$ by definition, whence $(z, z)\in Z$.

Let $(z_1, . . . , z_n)\in Z$. By definition of $c$, the conjugacy class $c(z_1, . . . , z_n)$ is equal to $c(z_n, z_1, . . . , z_{n-1})$ and is formed by the inverses of the elements of $c(\overline{z}_n, . . . ,\overline{z}_1)$. This shows that Z satisfies condition (ii).

With the notation of condition (iii), one can choose elements $u\in c(\mathbf{z}),v\in c(\mathbf{z}')$ and $w\in c(\mathbf{z}\mathbf{z}')$ such that $uv=w$. If two of the elements $u,v$ and $w$ belong to N, the same is true of the third, therefore N satisfies (iii).

The notation being that of (iv), we have, for every integer $i$ such that $1\leqslant i\leqslant n$, the relation

for $h$ is a homotopy joining $\varphi$ to $\psi$. This equality can also be written

Taking into account the relations $\widetilde{\psi}(f_i, \varepsilon_i) =\widetilde{\varphi}(f_{i+1}, \varepsilon_{i+1})$ for $1\leqslant i < n$ and $\widetilde{\psi}(f_n, \varepsilon_n) =\widetilde{\varphi}(f_1, \varepsilon_1)$, one deduces

so that the conjugacy classes $c((a_1, \varepsilon_1), . . . ,(a_n, \varepsilon_n))$ and $c((b_1, \varepsilon_1), . . . ,(b_n, \varepsilon_n))$ are equal. This shows that Z satisfies condition (iv) and completes the proof of the proposition.