Fixed points of nilpotent actions on S^sup 2

We prove that a nilpotent subgroup of orientation-preserving [formula omitted: see PDF] diffeomorphisms of [formula omitted: see PDF] has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation-preserving [formula omitted: see PDF] diffeomorphisms of [formula omitted: see PDF] preserving a compact set has a global fixed point. These results generalize theorems of Franks et al for the abelian case. We show that a nilpotent subgroup of orientation-preserving [formula omitted: see PDF] diffeomorphisms of [formula omitted: see PDF] 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 [formula omitted: see PDF] diffeomorphisms of [formula omitted: see PDF] .