The automorphism groups of the profinite braid groups

Automorphisms of procongruence curve and pants complexes

“…□ Remark 2.18. By Corollary C in [20] and Main Theorem in [10], there is actually a series of natural isomorphisms Out ♯ (P Γ(S 1,2 )) ∼ = Out ♯ (P Γ(S 0,5 )) ∼ = GT, where GT is the profinite Grothendieck-Teichmüller group, so that the monomorphism of Lemma 2.17 is actually an isomorphism. But we will not need this fact.…”