In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space \(\overline{{{\mathcal {T}}}}_g\) 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 \(\overline{{\mathcal {T}}}_g\).

在本文中，我们使用热带和对数几何的技术来构造Teichmüller 空间 的非阿基米德类比\(\overline{{{\mathcal {T}}}}_g\)其点是由在非阿基米德地区的稳定射影曲线和其Berkovich分析的拓扑基本群的Teichmüller标记组成的对。这种构造与 Gerritzen 和 Herrlich 对 Mumford 曲线的非阿基米德肖特基空间的经典构造密切相关并受其启发。我们认为非阿基米德 Teichmüller 空间的骨架正是 Chan-Melo-Viviani 引入的热带 Teichmüller 空间，作为 Culler-Vogtmann 外层空间的简单完成。因此，外层空间是\(\overline{{\mathcal {T}}}_g\)中平滑芒福德曲线轨迹的强烈变形收缩。