A non-Archimedean analogue of Teichmüller space and its tropicalization

<jats:title>Abstract</jats:title><jats:p>In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of<jats:italic>Teichmüller space</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\overline{{{\mathcal {T}}}}_g$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>T</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>g</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula>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<jats:inline-formula><jats:alternatives><jats:tex-math>$$\overline{{\mathcal {T}}}_g$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mover><mml:mi>T</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>g</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>