Skeletal filtrations of the fundamental group of a non-archimedean curve

Abstract

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first prove a simultaneous semistable reduction theorem for residually tame coverings, which we then use to construct a tropicalization functor from the category of residually tame coverings of a marked curve (X,D) to the category of tame coverings of a metrized complex Σ associated to (X,D). 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. We use this skeletal interpretation to define the absolute decomposition and inertia group of a curve, which can be seen as the first subgroups in a ramification filtration of the fundamental group of the curve. We prove that the cyclic coverings that arise from the corresponding decomposition and inertia quotients coincide with the coverings that arise from the toric and connected parts of the analytic Jacobian of the curve.