Proposition 8

Let $\delta$ be the canonical morphism of the free groupoid Grp(Γ) into G extending the canonical injective morphism of Γ into G. Let $a$ be a vertex of G and let A be the orbit of $a$ in G. For every $b\in A$, let $f_{ab}$ be an arrow of G joining $a$ to $b$. The isotropy group $H_a$ is the distinguished subgroup of $G_a$ generated by the elements Int($f_{ab}$)$(\delta (c))$, where $b$ runs through the set A and $c$ runs through the elements of Grp(Γ)$_b$.

If $x$ and $y$ are vertices of G belonging to the same orbit, with $x=\not a$, let us moreover fix an arrow $f_{xy}$ of G joining $x$ to $y$. For every point $x$ of G, let $N_x$ be the distinguished subgroup of $G_x$ generated by the elements Int($f_{xy}$)$(\delta (c))$, where $y$ runs through the orbit of $x$ in G and $c$ runs through the group Grp(Γ)$_y$. For every $f\in G_{xy}$ and every $g\in N_y$, one has $f gf^{-1}\in N_x$. Let $x\in$ Som(G). One has $\delta$(Grp(Γ)$_x$)$\subset N_x$, by definition of $N_x$. By definition of the groupoid H$,\delta (f)$ is an arrow of H for every arrow $f$ of Grp(Γ) ; consequently, $\delta$(Grp(Γ)$_x$) is contained in $H_x$. Since H is a distinguished subgroupoid of G, one then has

Int($f_{xy}$)$(\delta (c))\in$ Int($f_{xy}$)$(N_y)\subset$ Int($f_{xy}$)$(H_y)\subset H_x$

for every $y\in$ Som(G), and hence $N_x\subset H_x$.

Let $x$ and $y$ be vertices of Γ. Put $N_{x,y}=\emptyset$ if $x$ and $y$ do not belong to the same connected component of Γ. In the contrary case, let $c$ and $c'$ be classes of paths joining $x$ to $y$ in the quiver $\widetilde{\Gamma}$ associated with Γ. Let $z$ be a vertex of G belonging to the orbit of $y$ and let $\ell$ be an element of Grp(Γ)$_z$; in the group $G_x$, we have the equality:

The arrow $\delta (c(c')^{-1})$ is the class of a loop at $x$ in the quiver Γ, hence belongs to $N_x$, as does the arrow $f_{xz}\delta (\ell )f_{xz}^{-1}$. Since $\delta (c)f_{yz}f_{xz}^{-1}$ belongs to $G_x$ and $N_x$ is a distinguished subgroup of $G_x$, it follows that $\delta (c)f_{yz}\delta (\ell )f_{yz}^{-1}\delta (c')^{-1}$ belongs to $N_x$. Consequently, $\delta (c)N_y\subset N_x\delta (c')$. By symmetry, we have $\delta (c)N_y= N_x\delta (c')$. This implies that this set does not depend on the choice of $c$ and $c'$; we denote it by $N_{x,y}$.

Let N be the subquiver of G whose set of vertices is Som(G) and whose set of arrows joining $x$ to $y$ is equal to $N_{x,y}$ for every pair $(x, y)$ of vertices of G. It is a distinguished subgroupoid of G whose set of arrows contains F; hence H is a subgroupoid of N. In particular, $H_a\subset N_a$, whence the equality.