Soit une surface compacte connexe et soit un élément du groupe des homéomorphismes de isotopes à l’identité. Notons un relevé de au revêtement universel de . Fixons un domaine fondamental de ce revêtement universel. On dit que l’homéomorphisme est non-diffus si la suite converge vers , où désigne le diamètre de . Supposons que la surface est orientable de bord non vide. Nous démontrons que, si est distincte du disque et de l’anneau, un homéomorphisme est non-diffus si et seulement si il a des conjugués dans arbitrairement proches de l’identité. Dans le cas où la surface est l’anneau, nous démontrons qu’un homéomorphisme est non-diffus si et seulement si il a des conjugués arbitrairement proches d’une rotation (ce résultat était déjà connu dans la plupart des cas grâce à un théorème dû à Béguin, Crovisier, Le Roux et Patou). On en déduit que, pour de telles surfaces , un élément de est distordu si et seulement si il est non diffus.
Let be a compact connected surface and let be an element of the group of homeomorphisms of isotopic to the identity. Denote by a lift of to the universal cover of . Fix a fundamental domain of this universal cover. The homeomorphism is said to be non-spreading if the sequence converges to , where is the diameter of . Let us suppose now that the surface is orientable with a nonempty boundary. We prove that, if is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in arbitrarily close to the identity. In the case where the surface is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in arbitrarily close to a rotation (this was already known in most cases by a theorem by Béguin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces , an element of is distorted if and only if it is non-spreading.
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@article{JEP_2018__5__565_0,
author = {Emmanuel Militon},
title = {Conjugacy class of homeomorphisms and distortion elements in groups~of~homeomorphisms},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {565--604},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.78},
zbl = {06988588},
mrnumber = {3852261},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.78/}
}
TY - JOUR AU - Emmanuel Militon TI - Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 565 EP - 604 VL - 5 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.78/ DO - 10.5802/jep.78 LA - en ID - JEP_2018__5__565_0 ER -
%0 Journal Article %A Emmanuel Militon %T Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 565-604 %V 5 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.78/ %R 10.5802/jep.78 %G en %F JEP_2018__5__565_0
Emmanuel Militon. Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 565-604. doi : 10.5802/jep.78. https://jep.centre-mersenne.org/articles/10.5802/jep.78/
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