Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

The theories of uniformization for maximally degenerate curves over non-Archimedean curves (by Mumford) and for abelian varieties (by Raynaud) are one of the big achievements of modern arithmetic algebraic geometry. In recent years it has become clear that this story also has a tropical aspect. In fact, one may think of the construction as a two-step process : first construct a tropical uniformization, then use the combinatorial data of this tropical uniformization to build the non- Archimedean uniformization. In this talk, I will illustrate this principle in the case of the moduli space of curves. In particular I will explain how to build a non-Archimedean uniformization of the moduli space of stable algebraic curves that is closely connected to the non-Archimedean Schottky space for Mumford curves constructed by Gerritzen and Herrlich. Our approach will, in particular, exhibit tropical Teichmueller space, a simplicial compactification of Culler-Vogtmann Outer space, as a strong deformation retract of non-Archimedean Teichmueller space. The crucial technical ingredient will be the theory of Artin fans which will allow us to lift tropical data to algebraic moduli functors.