The notion of (G,X)-structure on a manifold introduced by Ehresman and popularized by Thurston is an interpretations of a "geometric structure" allowing via the developing map Theorem to efficiently express and prove uniformization results such as: "every compact locally Euclidean manifold is isomorphic to a quotient of the Euclidean space by a discrete group of isometry".
Singularities, such as conical singularities, are common place in constructions of manifolds endowed with such structures. On the one hand, the notion of singular (G,X)-manifold is not so well-defined and is usually understood as "there is a (G,X)-structure on the complement of the (n-2)-facet of a simplicial decomposition". On the other hand, we may want to use developing maps and holonomies to express and prove uniformization results for singular (G,X)- manifolds. Furthermore, leaving the realm of metric structures, interesting singularities become more complex.
In order to help manipulations of singular (G,X)-manifolds, we give a topological caracterization of reasonable singular locii and provide tools allowing to build developing maps in non-trivial situations. Doing so, we prove some new results on branched covering à la Fox.
To illustrate these methods we prove a uniformization result for flat Lorentzian 3-manifolds endowed with a type of cuspidal ends. | 
