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

Mann, Kathryn

doi:10.2140/agt.2012.12.421

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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

DOI: 10.2140/agt.2012.12.421

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 $r > (1 ∕ \sqrt{3}) k$ such that each point $x$ in the ball satisfies $∥ g (x) - h (x) ∥ \leq k$ 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 $(1 ∕ \sqrt{3}) k$ 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

References

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

Volume 12, issue 1 (2012)