---
_id: '66321'
abstract:
- lang: eng
  text: <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>
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Ulirsch M. A non-Archimedean analogue of Teichmüller space and its tropicalization.
    <i>Selecta Mathematica</i>. 2021;27(3):39. doi:<a href="https://doi.org/10.1007/s00029-021-00651-4">10.1007/s00029-021-00651-4</a>
  apa: Ulirsch, M. (2021). A non-Archimedean analogue of Teichmüller space and its
    tropicalization. <i>Selecta Mathematica</i>, <i>27</i>(3), 39. <a href="https://doi.org/10.1007/s00029-021-00651-4">https://doi.org/10.1007/s00029-021-00651-4</a>
  bibtex: '@article{Ulirsch_2021, title={A non-Archimedean analogue of Teichmüller
    space and its tropicalization}, volume={27}, DOI={<a href="https://doi.org/10.1007/s00029-021-00651-4">10.1007/s00029-021-00651-4</a>},
    number={3}, journal={Selecta Mathematica}, publisher={Springer Science and Business
    Media LLC}, author={Ulirsch, Martin}, year={2021}, pages={39} }'
  chicago: 'Ulirsch, Martin. “A Non-Archimedean Analogue of Teichmüller Space and
    Its Tropicalization.” <i>Selecta Mathematica</i> 27, no. 3 (2021): 39. <a href="https://doi.org/10.1007/s00029-021-00651-4">https://doi.org/10.1007/s00029-021-00651-4</a>.'
  ieee: 'M. Ulirsch, “A non-Archimedean analogue of Teichmüller space and its tropicalization,”
    <i>Selecta Mathematica</i>, vol. 27, no. 3, p. 39, 2021, doi: <a href="https://doi.org/10.1007/s00029-021-00651-4">10.1007/s00029-021-00651-4</a>.'
  mla: Ulirsch, Martin. “A Non-Archimedean Analogue of Teichmüller Space and Its Tropicalization.”
    <i>Selecta Mathematica</i>, vol. 27, no. 3, Springer Science and Business Media
    LLC, 2021, p. 39, doi:<a href="https://doi.org/10.1007/s00029-021-00651-4">10.1007/s00029-021-00651-4</a>.
  short: M. Ulirsch, Selecta Mathematica 27 (2021) 39.
date_created: 2026-07-08T06:48:52Z
date_updated: 2026-07-09T06:47:29Z
doi: 10.1007/s00029-021-00651-4
intvolume: '        27'
issue: '3'
language:
- iso: eng
page: '39'
publication: Selecta Mathematica
publication_identifier:
  issn:
  - 1022-1824
  - 1420-9020
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: A non-Archimedean analogue of Teichmüller space and its tropicalization
type: journal_article
user_id: '82981'
volume: 27
year: '2021'
...