Decompositions of tame profinite fundamental groups of non-archimedean curves using metrized complexes

In this paper, we study a natural covering functor from the category of tame etale coverings of a punctured curve over a complete algebraically closed non-archimedean field to the category of finite tame coverings of a metrized complex associated to the punctured curve. We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. This enhanced category of so-called rigidified tame coverings then inherits the structure of a Galois category, yielding a natural notion of a profinite fundamental group for metrized complexes. Using this graph-theoretical interpretation, we define the (absolute) decomposition and inertia groups of the metrized complex in the fundamental group of a nonpunctured curve and we show that they define normal subgroups. We then show that the quotient of the fundamental group by the decomposition group is isomorphic to the profinite completion of the ordinary fundamental group of the underlying graph of the metrized complex. Furthermore, we prove that the extensions that arise from the abelianization of the decomposition and inertia quotients coincide with the extensions that arise from the toric and connected parts of the analytic Jacobian of the curve.