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

TL;DR

This paper constructs a universal Mumford curve over the integers using Berkovich spaces, parametrizing Schottky groups and analyzing their uniformizations across all valued fields, linking algebraic geometry, group theory, and tropical geometry.

Contribution

It introduces a universal Mumford curve over $ ext{Spec}(Z)$ via a parameter space of Schottky groups, connecting non-archimedean and archimedean uniformizations.

Findings

Defined the analytic space $S_g$ parametrizing Schottky groups over all valued fields.

Proved the universal Mumford curve $C_g$ is uniformized by a universal Schottky group.

Described the action of $Out(F_g)$ on $S_g$ and related it to moduli spaces of tropical curves.

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.