Translator Disclaimer
2012 Bounded orbits and global fixed points for groups acting on the plane
Kathryn Mann
Algebr. Geom. Topol. 12(1): 421-433 (2012). DOI: 10.2140/agt.2012.12.421


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.


Download Citation

Kathryn Mann. "Bounded orbits and global fixed points for groups acting on the plane." Algebr. Geom. Topol. 12 (1) 421 - 433, 2012.


Received: 11 November 2011; Accepted: 18 November 2011; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1268.37067
MathSciNet: MR2916281
Digital Object Identifier: 10.2140/agt.2012.12.421

Primary: 37E30, 57M60

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 1 • 2012
Back to Top