A non-Archimedean analogue of Teichmüller space and its tropicalization

• Martin Ulirsch ^[1]
  1. [1] Goethe–Universität Frankfurt, Alemania
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 3, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00651-4
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• Resumen

  • In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space 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 T¯¯¯¯g.