Let be a group acting on by orientation-preserving homeomorphisms. We show that a tight bound on orbits implies a global fixed point. Precisely, if for some there is a ball of radius such that each point in the ball satisfies for all , and the action of satisfies a nonwandering hypothesis, then the action has a global fixed point. In particular any group of measure-preserving, orientation-preserving homeomorphisms of with uniformly bounded orbits has a global fixed point. The constant is sharp.
As an application, we also show that a group acting on by diffeomorphisms with orbits bounded as above is left orderable.
"Bounded orbits and global fixed points for groups acting on the plane." Algebr. Geom. Topol. 12 (1) 421 - 433, 2012. https://doi.org/10.2140/agt.2012.12.421