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

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.

v2 reflects published version. Added argument in section 2, results unchanged

References in corpus (1)

• Fixed Points of abelian actions on $S^2$

Cited by in corpus (1)

• Des points fixes communs pour des difféomorphismes de S^2 qui commutent et préservent une mesure de probabilité