Geometric features of Deligne-Mumford curves including hyperbolicity, affineness and inertia data can be detected from 3-step solvable quotients of their profinite fundamental groups, yielding partial anabelian reconstruction results for the curves and their rigidifications.
On Galois Action on the Inertia Stack of Moduli Spaces of Curves
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Anabelian geometry for Deligne-Mumford curves
Geometric features of Deligne-Mumford curves including hyperbolicity, affineness and inertia data can be detected from 3-step solvable quotients of their profinite fundamental groups, yielding partial anabelian reconstruction results for the curves and their rigidifications.