Fixed points of discrete nilpotent group actions on S^2

Abstract

We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 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.

15 pages, 2 figures