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In this paper we establish an essential property of the action of the GrothendieckTeichmuller group ĜT on the prime order torsion elements of the profinite fundamental group π 1 (M0,[n]). This property shows that the ĜT action on these torsion elements is of Galois type, as the absolute Galois action on geometric (discrete) inertia of π 1 (M0,[n]) is given by an analogous expression. The main difficulty in our study lies in characterising the profinite torsion conjugacy classes, which is an extremely difficult problem for profinite completions in general. In the groups π 1 (M0,[n]), we solve this problem for prime order torsion by showing that conjugacy classes are the same as the discrete ones. In what follows, we identify π 1 (M0,[n]) with the full mapping class group Γ0,[n], the group of oriented diffeomorphisms of the genus zero surface with n punctures modulo those isotopic to the identity.

Action of the Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups in genus zero