Fixed points of nilpotent actions on ${\mathbb{S}}^2$

Abstract

We prove that a nilpotent subgroup of orientation preserving $C^1$ diffeomorphisms of ${\mathbb{S}}^2$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^1$ diffeomorphisms of ${\mathbb{R}}^2$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case. We show that a nilpotent subgroup of orientation preserving $C^1$ diffeomorphisms of ${\mathbb{S}}^2$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover we study the properties of the two-points orbits of nilpotent fixed-point-free subgroups of orientation preserving $C^1$ diffeomorphisms of ${\mathbb{S}}^2$.

Some clarifications and minor corrections added

References in corpus (3)

• Sur les rapprochements par conjugaison en dimension 1 et classe C^1
• Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem
• Smoothing nilpotent actions on 1-manifolds

Cited by in corpus (4)

• Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem
• Fixed points of nilpotent actions on surfaces of negative Euler characteristic
• Distortion element in group of diffeomorphisms of the 2-sphere
• A note on fixed points of abelian actions in dimension one