Profinite completion of operads and the Grothendieck-Teichmüller group

In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little 2-disks is isomorphic to the profinite Grothendieck-Teichmüller group. In particular, the absolute Galois group of Q acts faithfully on the profinite completion of E2 in the homotopy category of profinite weak operads.