Uniformization of Deligne-Mumford curves

Abstract. 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 nd an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted pro-jective 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. 1