Fixed points of discrete nilpotent group actions on S²

الأصل · EN

We prove that for each integer k of at least 2, there is an open neigborhood νₖ of the identity map of the 2-sphere S², in C¹-topology such that: if G is a nilpotent subgroup of Diff¹(S²) with length k of nilpotency, generated by elements in νₖ, then the natural action on S² has non-empty fixed point set. Moreover, the G-action has at least two fixed points if the action has a finite non-trivial orbit.