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

Abstract

For every integer $g \ge 1$ 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 $\hbox {Out}(F_g)$ of outer automorphisms of the free group with g generators on ${{\mathcal {S}}}_g$, describing the quotient $\hbox {Out}(F_g) \backslash {{\mathcal {S}}}_g$ 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.