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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

Abstract

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.

Citation

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Kathryn Mann. "Bounded orbits and global fixed points for groups acting on the plane." Algebr. Geom. Topol. 12 (1) 421 - 433, 2012. https://doi.org/10.2140/agt.2012.12.421

Information

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

Subjects:
Primary: 37E30, 57M60

Rights: Copyright © 2012 Mathematical Sciences Publishers

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