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.
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