[Un horocycle divergent dans le compactifié par horofonctions de la métrique de Teichmüller]
Nous donnons un exemple d’horocycle dand l’espace de Teichmüller de la sphère épointée cinq fois qui ne converge pas dans le compactifié de Gardiner–Masur, ou, de façon équivalente, dans le compactifié par horofonctions de la métrique de Teichmüller. Comme étape intermédiaire, nous exhibons une courbe simple fermée dont la longueur extrémale est périodique mais pas constante le long de l’horocycle. L’exemple se relève à tout espace de Teichmüller de dimension complexe supérieure à un par des constructions de revêtement.
We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner–Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichmüller space of complex dimension greater than one via covering constructions.
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Keywords: Horocycle, horofunction, Teichmüller space, extremal length.
Mot clés : Horocycle, horofonction, espace de Teichmüller, longueur extrémale
Fortier Bourque, Maxime 1
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@article{AIF_2023__73_5_1885_0,
author = {Fortier Bourque, Maxime},
title = {A divergent horocycle in the horofunction compactification of the {Teichm\"uller} metric},
journal = {Annales de l'Institut Fourier},
pages = {1885--1902},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {73},
number = {5},
year = {2023},
doi = {10.5802/aif.3564},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3564/}
}
TY - JOUR AU - Fortier Bourque, Maxime TI - A divergent horocycle in the horofunction compactification of the Teichmüller metric JO - Annales de l'Institut Fourier PY - 2023 SP - 1885 EP - 1902 VL - 73 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3564/ DO - 10.5802/aif.3564 LA - en ID - AIF_2023__73_5_1885_0 ER -
%0 Journal Article %A Fortier Bourque, Maxime %T A divergent horocycle in the horofunction compactification of the Teichmüller metric %J Annales de l'Institut Fourier %D 2023 %P 1885-1902 %V 73 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3564/ %R 10.5802/aif.3564 %G en %F AIF_2023__73_5_1885_0
Fortier Bourque, Maxime. A divergent horocycle in the horofunction compactification of the Teichmüller metric. Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 1885-1902. doi : 10.5802/aif.3564. https://aif.centre-mersenne.org/articles/10.5802/aif.3564/
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