Area preserving group actions on S 2

The main result of this paper is that every action of a finite index subgroup of SL(3, Z) on S2 by area preserving diffeomorphisms factors through a finite group. An important tool we use, which may be of independent interest is the result that if F : S2 → S2 is a non-trivial, area preserving, orientation preserving diffeomorphism and if Fix(F ) contains at least three points, then F has points of arbitrarily high period. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism F : S → S of a closed orientable surface has a normal form. If the fixed point set is finite this is just the Thurston normal form.