In the theory of algebraic curves, a prominent role is played by uniformization, the description of a curve as the quotient of an open subset of the projective line by the action of a discrete group. All complex algebraic curves admit uniformization, as well as some non-archimedean algebraic curves, called Mumford curves.
In this talk, I present a construction of "universal Mumford curves", spaces that parametrize both archimedean and non-archimedean uniformizable curves, using the theory of analytic geometry over $\mathbb Z$. These spaces are tightly related to moduli spaces of tropical curves, in such a way that their arithmetic is encoded by combinatorial invariants. Understanding these latter can give important information on how uniformization properties behave "in families" of curves. As an instance of this phenomenon, I will highlight the presence of canonical $\Delta$-complexes appearing in the construction of universal Mumford curves.Finally, I will say a few words about how to use them to compute the cohomology of the moduli space of curves and to study the geometry of certain character varieties.
This is joint work with Jérôme Poineau.