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

TL;DR

This paper proves that nilpotent groups of orientation-preserving $C^{1}$ diffeomorphisms on ${ m S}^2$ and ${ m R}^2$ have finite orbits or fixed points, extending previous results for abelian groups and analyzing orbit properties.

Contribution

The paper generalizes known theorems for abelian groups to nilpotent groups, establishing existence of fixed points and finite orbits for such groups acting on ${ m S}^2$ and ${ m R}^2$.

Findings

Nilpotent subgroups of ${ m S}^2$ have finite orbits of size at most two.

Finitely generated nilpotent subgroups of ${ m R}^2$ with invariant compact sets have fixed points.

Nilpotent subgroups with odd finite orbits on ${ m S}^2$ have fixed points.

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 ${\mathbb S}^{2}$.