Algebraic & Geometric Topology

Bounded orbits and global fixed points for groups acting on the plane

Kathryn Mann

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Let G be a group acting on 2 by orientation-preserving homeomorphisms. We show that a tight bound on orbits implies a global fixed point. Precisely, if for some k>0 there is a ball of radius r>(13)k such that each point x in the ball satisfies g(x)h(x)k for all g,hG, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed point. In particular any group of measure-preserving, orientation-preserving homeomorphisms of 2 with uniformly bounded orbits has a global fixed point. The constant (13)k is sharp.

As an application, we also show that a group acting on 2 by diffeomorphisms with orbits bounded as above is left orderable.

Article information

Algebr. Geom. Topol., Volume 12, Number 1 (2012), 421-433.

Received: 11 November 2011
Accepted: 18 November 2011
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 57M60: Group actions in low dimensions

fixed point planar action group action prime end left order plane homeomorphism Brouwer plane translation


Mann, Kathryn. Bounded orbits and global fixed points for groups acting on the plane. Algebr. Geom. Topol. 12 (2012), no. 1, 421--433. doi:10.2140/agt.2012.12.421.

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  • M Barge, R M Gillette, A fixed point theorem for plane separating continua, Topology Appl. 43 (1992) 203–212
  • R G Burns, V W D Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972) 441–445
  • D Calegari, Circular groups, planar groups, and the Euler class, from: “Proceedings of the Casson Fest”, (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 431–491
  • C Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913) 323–370
  • J Franks, M Handel, K Parwani, Fixed points of abelian actions on $S\sp 2$, Ergodic Theory Dynam. Systems 27 (2007) 1557–1581
  • J Franks, P Le Calvez, Regions of instability for non-twist maps, Ergodic Theory Dynam. Systems 23 (2003) 111–141
  • J N Mather, Topological proofs of some purely topological consequences of Carathéodory's theory of prime ends, from: “Selected studies: physics-astrophysics, mathematics, history of science”, (T M Rassias, G M Rassias, editors), North-Holland, Amsterdam (1982) 225–255
  • C Pommerenke, Univalent functions, Studia Math. Lehrbücher XXV, Vandenhoeck & Ruprecht, Göttingen (1975) With a chapter on quadratic differentials by G Jensen
  • W P Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974) 347–352
  • D Witte, Arithmetic groups of higher ${\bf Q}$–rank cannot act on $1$–manifolds, Proc. Amer. Math. Soc. 122 (1994) 333–340