# A non-Archimedean analogue of Teichmüller space and its tropicalization

Authors
• 1 Goethe–Universität Frankfurt, Frankfurt am Main, 60325, Germany , Frankfurt am Main (Germany)
Type
Published Article
Journal
Selecta Mathematica
Publisher
Springer International Publishing
Publication Date
May 29, 2021
Volume
27
Issue
3
Identifiers
DOI: 10.1007/s00029-021-00651-4
Source
Springer Nature
Keywords
In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller spaceT¯g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{{\mathcal {T}}}}_g$$\end{document} 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 T¯g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\mathcal {T}}}_g$$\end{document}.