Operads of genus zero curves and the Grothendieck–Teichmüller group

Abstract

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck–Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck–Teichmüller group acts nontrivially on ${\overline{\mathcal{ℳ}}}_{0,\bullet +1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little $2$–disks operad is formal.