Fixed points of discrete nilpotent group actions on S 2

Suely Druck; Fuquan Fang[1]; Sebastião Firmo[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 (2002)

  • Volume: 52, Issue: 4, page 1075-1091
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Druck, Suely, Fang, Fuquan, and Firmo, Sebastião. "Fixed points of discrete nilpotent group actions on $S^2$." Annales de l’institut Fourier 52.4 (2002): 1075-1091. <http://eudml.org/doc/116004>.

@article{Druck2002,
abstract = {We prove that for each integer $k\ge 2$ there is an open neighborhood $\{\mathcal \{V\}\}_k$ of the identity map of the 2-sphere $S^2$, in $C^1$ topology such that: if $G$ is a nilpotent subgroup of $\{\rm Diff\}^1(S^2)$ with length $k$ of nilpotency, generated by elements in $\{\mathcal \{V\}\}_k$, 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.},
affiliation = {Universidade Federal Fluminense, Instituto de Matematica, Rua Mário Santos Braga s/n, Valonguinho, 24020-140 Niterói RJ (Brésil); 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)},
author = {Druck, Suely, Fang, Fuquan, Firmo, Sebastião},
journal = {Annales de l’institut Fourier},
keywords = {group action; nilpotent group; fixed point; diffeomorphisms of },
language = {eng},
number = {4},
pages = {1075-1091},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fixed points of discrete nilpotent group actions on $S^2$},
url = {http://eudml.org/doc/116004},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Druck, Suely
AU - Fang, Fuquan
AU - Firmo, Sebastião
TI - Fixed points of discrete nilpotent group actions on $S^2$
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 1075
EP - 1091
AB - We prove that for each integer $k\ge 2$ there is an open neighborhood ${\mathcal {V}}_k$ of the identity map of the 2-sphere $S^2$, in $C^1$ topology such that: if $G$ is a nilpotent subgroup of ${\rm Diff}^1(S^2)$ with length $k$ of nilpotency, generated by elements in ${\mathcal {V}}_k$, 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.
LA - eng
KW - group action; nilpotent group; fixed point; diffeomorphisms of
UR - http://eudml.org/doc/116004
ER -

References

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  10. E. Lima, Commuting vector fields on S 2 , Proc. Amer. Math. Soc. 15 (1964), 138-141 Zbl0117.17002MR159342
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  13. H. Poincaré, Sur les courbes définis par une équation différentielle, J. Math. Pures Appl. 4 (1885), 167-244 
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  15. J. Rotman, An introduction to the theory of groups, (1995), Springer-Verlag Zbl0810.20001MR1307623

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