Corollary 1

Under the assumptions of proposition 1, suppose moreover that, for every $i\in$ I, the image of the homomorphism from $\pi_1(A_i, a(i))$ into $\pi_1(Y, a(i))$, induced by the canonical injection of $A_i$ into Y, is trivial. The homomorphism $\mathsf{M}': F(J)\rightarrow \pi_1(Y, a(i_0))$ induced by $\mathsf{M}$ by restriction is then surjective, and its kernel is the smallest distinguished subgroup containing the elements $j\in$ Fl(T) and the elements $j_{12}(k)j_{23}(k)j_{13}(k)^{-1}$ for $k\in K$.

Let $\pi : (_i*_{\in I}\pi_1(A_i, a(i)))*F(J)\rightarrow F(J)$ be the unique homomorphism which induces the trivial homomorphism on each $\pi_1(A_i, a(i))$ and the identity on F(J); it is surjective. Let $i\in I$. The definition of $\mathsf{M}$ and the assumption that the image of the homomorphism from $\pi_1(A_i, a(i))$ into $\pi_1(Y, a(i))$ induced by the canonical injection is trivial imply that, for every $v\in \pi_1(A_i, a(i)),\mathsf{M}(v)$ is the identity element of $\pi_1(Y, a(i_0))$. Hence $\mathsf{M}=\mathsf{M}'\circ \pi$. It follows that the homomorphism $\mathsf{M}'$ is surjective, and its kernel is the smallest distinguished subgroup containing the images under $\pi$ of the elements $r_1(j),r_2(j, v)$ and $r_3(k)$ defined in Prop. 1. For $j\in$ Fl(T), we have $\pi (r_1(j)) =j$. For every $j\in$ J and every $v\in \pi_1(V, b(j))$, we have $\pi (r_2(j, v)) =e$. Finally, for every $k= (i_1, i_2, i_3,U)\in K$ and every $s\in$ $\{1,2,3\},\lambda_s(k)$ is the class of a loop in $A_{i_s}$; hence $\pi (\lambda_s(k)) =e$, so that $\pi (r_3(k)) =j_{12}(k)j_{23}(k)j_{13}(k)^{-1}$. The corollary follows.