Anabelian geometry for Deligne-Mumford curves

TL;DR

This paper develops an anabelian framework for Deligne-Mumford curves, showing their geometric structures can be reconstructed from low-level quotients of their profinite fundamental groups, advancing understanding of their group-theoretic properties.

Contribution

It introduces a novel anabelian approach for Deligne-Mumford curves, demonstrating that key geometric features are detectable from low-level solvable quotients of their fundamental groups.

Findings

Hyperbolicity, affineness, and inertia data are detectable from 3-step quotients.

Established a 5-step anabelian theorem for affine Deligne-Mumford curves.

Reconstructed Deligne-Mumford curves and their variants from fundamental group data.

Abstract

We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their \'etale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups.