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

TL;DR

This paper proves that certain groups acting on the plane with bounded orbits and a nonwandering condition must have a global fixed point, establishing sharp bounds and implications for group orderability.

Contribution

It introduces sharp bounds for fixed points in plane group actions and links bounded orbits to group orderability under specific conditions.

Findings

Groups with bounded orbits have a global fixed point.

The bound k/√3 is optimal.

Such groups are left orderable.

Abstract

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < 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 the plane with uniformly bounded orbits has a global fixed point. The constant k/\sqrt{3} is sharp. We also show that a group acting on the plane with orbits bounded as above is left orderable.