Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T04:30:31.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed points of abelian actions on S2

Published online by Cambridge University Press:  01 October 2007

JOHN FRANKS ,
MICHAEL HANDEL  and
KAMLESH PARWANI
JOHN FRANKS
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA (email: [email protected])
MICHAEL HANDEL
Affiliation:
Department of Mathematics and Computer Science, Herbert H Lehman College (CUNY), 250, Bedford Park, Boulevard West, Bronx, NY 10468, USA (email: [email protected])
KAMLESH PARWANI
Affiliation:
Department of Mathematics, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL 61920-3099, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that if is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of which leaves invariant a compact set then there is a common fixed point for all elements of . We also show that if is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of with index at most two.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1557 - 1581
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bonatti, C.. Un point fixe common pour des diffeomorphisms commutants de S 2. Ann. of Math. (2) 129 (1989), 61–79.CrossRefGoogle Scholar
[2]Brown, R. F.. The Lefschetz Fixed Point Theorem. Scott, Foresman and Co., Glenview, IL, 1971.Google Scholar
[3]Calegari, D.. Circular groups, planar groups, and the Euler class. Proc. Casson Fest (Geometry and Topology Monographs, 7). 2004, pp. 431–491.Google Scholar
[4]Dold, A.. Fixed point index and fixed point theorem for Euclidean neighborhood retracts. Topology 4 (1965), 1–8.CrossRefGoogle Scholar
[5]Druck, S., Fang, F. and Firmo, S.. Fixed points of discrete nilpotent group actions on S 2. Ann. Inst. Fourier (Grenoble) 52(4) (2002), 1075–1091.CrossRefGoogle Scholar
[6]Fathi, A., Laudenbach, F. and Poenaru, V.. Travaux de Thurston sur les surfaces. Astérisque (1979), 66–67.Google Scholar
[7]Franks, J. and Handel, M.. Periodic points of Hamiltonian surface diffeomorphisms. Geom. Topol. 7 (2003), 713–756.CrossRefGoogle Scholar
[8]Gambaudo, J.-M.. Periodic orbits and fixed points of a C 1 orientation-preserving embedding of D 2. Math. Proc. Cambridge Philos. Soc. 108 (1990), 307–310.CrossRefGoogle Scholar
[9]Handel, M.. Commuting homeomorphisms of S 2. Topology 31 (1992), 293–303.CrossRefGoogle Scholar
[10]Handel, M.. A fixed point theorem for planar homeomorphisms. Topology 38 (1999), 235–264.CrossRefGoogle Scholar
[11]Handel, M. and Thurston, W.. New proofs of some results of Nielsen. Adv. Math. 56 (1985), 173–191.CrossRefGoogle Scholar
[12]Hirsch, M.. Common fixed points for two commuting surface homeomorphisms. Houston J. Math. 29(4) (2003), 961–981.Google Scholar
[13]Kerchoff, S.. The Nielsen realization problem. Ann. of Math. (2) 117 (1983), 235–265.CrossRefGoogle Scholar
[14]Kneser, H.. Die deformationssatze der einfach zusammenhangenden flacher. Math. Z. 25 (1926), 362–372.CrossRefGoogle Scholar
[15]Thurston, W. P.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.CrossRefGoogle Scholar
[16]Warner, F.. Foundations of Differentiable Manifolds and Lie Groups. Scott Foresman and Co., Glenview, IL, 1971.Google Scholar