Volume 12, issue 1 (2012)

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

Kathryn Mann

Algebraic & Geometric Topology 12 (2012) 421–433
Abstract

Let $G$ be a group acting on ${ℝ}^{2}$ 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 $r>\left(1∕\phantom{\rule{0.3em}{0ex}}\sqrt{3}\right)k$ such that each point $x$ in the ball satisfies $\parallel g\left(x\right)-h\left(x\right)\parallel \le k$ for all $g,h\in G$, 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 ${ℝ}^{2}$ with uniformly bounded orbits has a global fixed point. The constant $\left(1∕\phantom{\rule{0.3em}{0ex}}\sqrt{3}\right)k$ is sharp.

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

Keywords
fixed point, planar action, group action, prime end, left order, plane homeomorphism, Brouwer plane translation
Mathematical Subject Classification 2010
Primary: 37E30, 57M60
Publication
Received: 11 November 2011
Accepted: 18 November 2011
Published: 20 March 2012
Authors
 Kathryn Mann Department of Mathematics University of Chicago 5734 University Ave Chicago IL 60637 USA http://math.uchicago.edu/~mann