Nature of the flat/sharp quasi-special condition on automorphisms of profinite mapping class groups

Determine whether the quasi-special and symmetric conditions (denoted flat/sharp) imposed on automorphisms of the profinite mapping class groups \widehat{\Gamma}_{0,m} of the genus-zero moduli spaces M_{0,m} are of arithmetic origin or purely group-theoretic. Specifically, ascertain whether the requirements that an automorphism (i) preserve the conjugacy classes \mathfrak{X}_{ij} of cuspidal inertia subgroups generated by loops around the boundary divisors D_{ij} in the Deligne–Mumford compactification \overline{M}^{DM}_{0,[m]}, and (ii) commute with the natural S_m-action on marked points, can be characterized intrinsically within group theory independent of Galois-theoretic input.

Background

In Section 2, the paper introduces quasi-special automorphisms of the geometric étale fundamental group (\widehat{\Gamma}{0,m}) that preserve the conjugacy classes of cuspidal inertia subgroups associated to boundary divisors D{ij} (the “flat” condition), and, to reflect the geometry of M_{0,[m]}, further considers automorphisms that commute with the symmetric group action on marked points (the “sharp” condition). These conditions are used to define a subgroup of automorphisms that enables a stable identification with the Grothendieck–Teichmüller group GT across the modular tower.

While the braid-theoretic and anabelian constructions motivate these conditions, the paper explicitly notes that the conceptual source of the flat/sharp constraints—whether they arise essentially from arithmetic (Galois) considerations or can be justified purely within group theory—was unclear at that stage, prompting the following open question.