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

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 $r>\left(1∕\sqrt{3}\right)k$ 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 $\left(1∕\sqrt{3}\right)k$ 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.