Fixed points of discrete nilpotent group actions on S 2
Annales de l'Institut Fourier, Volume 52 (2002) no. 4, p. 1075-1091

We prove that for each integer k2 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 k2 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 : https://doi.org/10.5802/aif.1912
Classification:  37B05,  37C25,  37C85
Keywords: group action, nilpotent group, fixed point
@article{AIF_2002__52_4_1075_0,
     author = {Druck, Suely and Fang, Fuquan and Firmo, Sebasti\~ao},
     title = {Fixed points of discrete nilpotent group actions on $S^2$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     pages = {1075-1091},
     doi = {10.5802/aif.1912},
     zbl = {1005.37019},
     mrnumber = {1926674},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_4_1075_0}
}
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. http://www.numdam.org/item/AIF_2002__52_4_1075_0/

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