Schottky spaces and universal Mumford curves over $\mathbb{Z}$

Abstract: For every integer $g \geq 1$ we define a universal Mumford curve of genus $g$ in the framework of Berkovich spaces over $\mathbb{Z}$. This is achieved in two steps: first, we build an analytic space $\mathcal{S}g$ that parametrizes marked Schottky groups over all valued fields. We show that $\mathcal{S}_g$ is an open, connected analytic space over $\mathbb{Z}$. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of $\mathcal{S}_g$, both locally and globally. As a result, we can define the universal Mumford curve $\mathcal{C}_g$ as a relative curve over $\mathcal{S}_g$ such that every Schottky uniformized curve can be described as a fiber of a point in $\mathcal{S}_g$. We prove that the curve $\mathcal{C}_g$ is itself uniformized by a universal Schottky group acting on the relative projective line $\mathbb{P}^1{\mathcal{S}_g}$. Finally, we study the action of the group $Out(F_g)$ of outer automorphisms of the free group with $g$ generators on $\mathcal{S}_g$, describing the quotient $Out(F_g) \backslash \mathcal{S}_g$ in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.