Algebraic & Geometric Topology

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

Kathryn Mann

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715344

Digital Object Identifier
doi:10.2140/agt.2012.12.421

Mathematical Reviews number (MathSciNet)
MR2916281

Zentralblatt MATH identifier
1268.37067

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

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

Citation

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. https://projecteuclid.org/euclid.agt/1513715344


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