Fixed points of discrete nilpotent group actions on $S^2$

Druck, Suely; Fang, Fuquan; Firmo, Sebastião

doi:10.5802/aif.1912

Fixed points of discrete nilpotent group actions on

S^{2}

[Points fixes d’actions des groupes nilpotents discrets sur

S^{2}

]

Druck, Suely ; Fang, Fuquan ¹ ; Firmo, Sebastião ²

¹ Universidade Federal Fluminense, Instituto de Matematica, Rua Mário Santos Braga s/n, Valonguinho, 24020-140 Niterói RJ (Brésil)
² Nankai Institute of Mathematics, Tianjin 300071 (Rép. Pop. Chine) et Universidade Federal Fluminense, Instituto de Matematica, Rua Mário Santos Braga s/n, Valonguinho, 24020-140 Niterói RJ (Brésil)

Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1075-1091.

Résumé
Abstract

On démontre que pour chaque entier $k \geq 2$ il existe un voisinage ouvert $𝒱_{k}$ de l’application identité de la 2-sphère, pour la $C^{1}$ topologie, tel que : si $G \subset {Diff}^{1} (S^{2})$ est un sous-groupe nilpotent à longueur de nilpotence $k$ , engendré par une famille quelconque d’éléments de $𝒱_{k}$ , alors l’action naturelle de $G$ sur $S^{2}$ a un point fixe. De plus, en présence d’une orbite finie cette action a au moins deux points fixes.

We prove that for each integer $k \geq 2$ there is an open neighborhood $𝒱_{k}$ of the identity map of the 2-sphere $S^{2}$ , in $C^{1}$ topology such that: if $G$ is a nilpotent subgroup of ${Diff}^{1} (S^{2})$ with length $k$ of nilpotency, generated by elements in $𝒱_{k}$ , then the natural $G$ -action on $S^{2}$ has nonempty fixed point set. Moreover, the $G$ -action has at least two fixed points if the action has a finite nontrivial orbit.

MR Zbl

DOI : 10.5802/aif.1912

Classification : 37B05, 37C25, 37C85
Keywords: group action, nilpotent group, fixed point
Mot clés : action de groupe, groupe nilpotent, point fixe

Affiliations des auteurs :

Druck, Suely ; Fang, Fuquan ¹ ; Firmo, Sebastião ²

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     title = {Fixed points of discrete nilpotent group actions on $S^2$},
     journal = {Annales de l'Institut Fourier},
     pages = {1075--1091},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     mrnumber = {1926674},
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Druck, Suely; Fang, Fuquan; Firmo, Sebastião. Fixed points of discrete nilpotent group actions on $S^2$. Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1075-1091. doi : 10.5802/aif.1912. https://www.numdam.org/articles/10.5802/aif.1912/

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