A non-Archimedean analogue of Teichmüller space and its tropicalization

  • 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.
Author:Martin UlirschORCiDGND
Parent Title (English):Selecta Mathematica
Place of publication:Basel [u.a.] ; Berlin
Document Type:Article
Date of Publication (online):2021/05/29
Date of first Publication:2021/05/29
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2022/07/12
Issue:3, art. 39
Page Number:34
First Page:1
Last Page:34
Finally, we acknowledge support from the LOEWE-Schwerpunkt “Uniformisierte Strukturen in Arithmetik und Geometrie”. Open Access funding enabled and organized by Projekt DEAL.
Weitere MSC-Klassifikation: 14T20.
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:14-XX ALGEBRAIC GEOMETRY / 14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] / 14G22 Rigid analytic geometry
14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H10 Families, moduli (algebraic)
Licence (German):License LogoCreative Commons - Namensnennung 4.0