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.
A Series of Modern Surveys in Mathematics
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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.