The theory 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 curves. In particular I will explain how to build a non-Archimedean uniformization of the moduli space of algebraic curves that compactifies the non-Archimedean Teichmueller 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 Gerritzen-Herrlich-Teichmueller space. This is based on joint work in progress with A. Werner.

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