Proposition 9

Let $a$ be a vertex of G. There exists a unique homomorphism $\lambda$ of the free group F(S-Fl(A)) into $\pi_1(G, a)$ such that

for $f\in S$ - Fl(A). The homomorphism $\lambda$ is a group isomorphism.

Let us denote by L the group F(S-Fl(A)). The existence and uniqueness of the homomorphism $\lambda : L\rightarrow \pi_1(G, a)$ satisfying relations (1) follows from A, I, p. 85, prop. 8. Let $\mathscr{L}$ be the groupoid associated with the group L (II, p. 163, example 1); let us denote by $s$ its unique vertex. There exists a unique groupoid morphism $\mu:$ Grp(G) $\rightarrow \mathscr{L}$ such that $\mu(\theta_G(f)) =f$ for every arrow $f\in$ S-Fl(A) and $\mu(\theta_G(f)) =e_s$ for every arrow $f\in$ Fl(A) (II, p. 173, prop. 5). For $f\in$ S-Fl(A), one has $\mu(\theta_G(f)) =f$; the homomorphism $\mu_a\circ \lambda$ is therefore the identity isomorphism of the group L (A, I, p. 85, prop. 8). It follows that $\lambda$ is injective.

Let $\varpi_G/A$ be the groupoid deduced from $\varpi_G$ by contraction of the arrows of A and let $p:\varpi_G\rightarrow \varpi_G/A$ be the canonical groupoid morphism. It is surjective and the group homomorphism $p_a$ deduced from $p$ by passing to the isotropy groups is an isomorphism (II, p. 178, cor. 2 of prop. 8). Since the graph G is assumed connected and A is a maximal tree in it, the groupoid $\varpi_G/A$ has a unique vertex (II, p. 176). Moreover, $\varpi_G$ is generated by $\theta_G(G)$; consequently, $\varpi_G/A$ is generated by the loops $p(f)$, for $f\in S$, and the group $(\varpi_G/A)_{p(a)}$ is generated by the elements of the form $p(\theta_G(f))$, for $f\in$ S-Fl(A). The homomorphism $p_a\circ \lambda$ is therefore surjective. Consequently, $\lambda$ is surjective.