no. 4
Fixed points of discrete nilpotent group actions on S 2
Druck, Suely; Fang, Fuquan1; Firmo, Sebastião2
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, Volume 52 (2002) no. 4, pp. 1075-1091.

We prove that for each integer k≥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.

On démontre que pour chaque entier k≥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⊂ 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.

DOI: 10.5802/aif.1912
Classification: 37B05, 37C25, 37C85
Keywords: group action, nilpotent group, fixed point
Keywords: action de groupe, groupe nilpotent, point fixe
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) 
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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, Volume 52 (2002) no. 4, pp. 1075-1091. doi : 10.5802/aif.1912. https://aif.centre-mersenne.org/articles/10.5802/aif.1912/

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