A non-Archimedean analogue of Teichm\"uller space and its tropicalization

In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichm"uller space ${\bar{\mathcal{T}}}_g$ whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichm"uller 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 Teichm"uller space is precisely the tropical Teichm"uller 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 ${\bar{\mathcal{T}}}_g$.