Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. In this article we define the group $\operatorname{Out}^{\sharp }_{n}$ of quasi-special symmetric outer automorphisms of the algebraic fundamental group $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ for all $n\ge 4$ to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ and commuting with the group of outer automorphisms of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ obtained by permuting the marked points. Our main result states that $\operatorname{Out}^{\sharp }_{n}$ is isomorphic to the Grothendieck-Teichmüller group $\widehat {\operatorname{GT}}$for all $n\ge 5$.