Abstract
In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller spaceoverline{{{mathcal {T}}}}_g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in overline{{mathcal {T}}}_g.
Highlights
Let g 2 and suppose for that we are working over C
Teichmüller space Tg,C is the universal cover of the moduli space Mg of smooth curves of genus g
A candidate for a tropical analogue of Teichmüller space is Outer space CVg in the sense of Culler–Vogtmann, which arrived in the world of mathematics well before the recent spark in interest in tropical geometry
Summary
Let g 2 and suppose for that we are working over C. It is a complex analytic space that functions as a fine moduli space of smooth curves X (of genus g) together with a Teichmüller marking, that is a an equivalence φ : π1(X) −→∼ Πg1, where
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