Fundamental groups of moduli and the Grothendieck-Teichmüller group

Abstract

Let Mo,, denote the moduli space of Riemann spheres with n ordered marked points. In this article we define the group Out# of quasispecial symmetric outer automorphisms of the algebraic fundamental group 1r1(Mo,n) for all n > 4 to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of Irl(M4,n) and commuting with the group of outer automorphisms of rl (MO,n) obtained by permuting the marked points. Our main result states that Outd is isomorphic to the Grothendieck-Teichmiiller group GT for all n > 5.