## and B.Noohi, Uniformization of Deligne-Mumford curves

Citations: | 7 - 0 self |

### BibTeX

@MISC{Behrend_andb.noohi,,

author = {Kai Behrend and Behrang Noohi},

title = {and B.Noohi, Uniformization of Deligne-Mumford curves},

year = {}

}

### OpenURL

### Abstract

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 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. 1.

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Citation Context ...: 2-groups In this appendix we gather some results on 2-groups and crossed-modules that are used throughout the paper. Our main reference is [8]. More can be found in [4] (and references therein) and =-=[1]-=-. Quick recall on 2-groups. A 2-group G is a group object in the category of groupoids. Alternatively, we can define a 2-group to be a groupoid object in the category of groups, or also, as a (strict)... |

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Citation Context ... passing to a quotient, in which case the statement is again obvious. Case C: g = 0, l = 0 and k ≥ 3. Again, by passing to a quotient, we may assume k = 3. For any triple of integers m, n, p ≥ 2, Fox =-=[2]-=- constructs two permutations a, b in a certain finite permutation group such that a has order m, b has order n, and ab has order p. This can be used to construct the desired quotient (also see Remark ... |

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Citation Context ... as saying that the morphism G → AutX is “injective”. That is, its kernel is equivalent to the trivial 2-group. Proof of Theorem 8.1. Part (i) is a GAGA type statement and follows from Theorem 1.1 of =-=[5]-=-. To prove part (ii) we may assume r ≥ 2. By the previous lemma, the map PGL(n1, n2, · · · , nr) → Aut P(n1, n2, · · · , nr) induces an isomorphism on π2 and an injection on π1. So we only need to pro... |