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

Abstract

For every integer g ≥ 1 we define a universal Mumford curve of genus g in the framework of Berkovich spaces over Z. This is achieved in two steps: first, we build an analytic space Sg that parametrizes marked Schottky groups over all valued fields. We show that Sg is an open, connected analytic space over Z. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of Sg, both locally and globally. As a result, we can define the universal Mumford curve Cg as a relative curve over Sg such that every Schottky uniformized curve can be described as a fiber of a point in Sg. We prove that the curve Cg is itself uniformized by a universal Schottky group acting on the relative projective line P 1Sg . Finally, we study the action of the group Out(Fg) of outer automorphisms of the free group with g generators on Sg, describing the quotient Out(Fg)\Sg 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.