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

Let G be a group acting on R 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 √ 3 k such that each point x in the ball satisfies ‖g(x) − h(x)‖ ≤ k for all g, h ∈ 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 with uniformly bounded orbits has a global fixed point. The constant 1 √ 3 k is sharp. As an application, we also show that a group acting on R by diffeomorphisms with orbits bounded as above is left orderable. AMS Classification: 37E30, 57M60