Fixed points of nilpotent actions on

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 Frankset al 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 2-points orbits of nilpotent fixed-point-free subgroups of orientation-preserving$C^{1}$diffeomorphisms of$\mathbb{S}^{2}$.