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 ${\mathbb{P}}_{{\mathcal{S}}_g}^1$. 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.

40 pages, 2 figures. Comments welcome

References in corpus (1)

• La structure des courbes analytiques