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Uniformization of Deligne-Mumford curves

2006; De Gruyter; Volume: 2006; Issue: 599 Linguagem: Inglês

10.1515/crelle.2006.080

1435-5345

Kai Behrend, Behrang Noohi,

Geometric and Algebraic Topology

We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks 𝒫(n1, n2, … nr). We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups.