SOME VIRTUALLY ABELIAN SUBGROUPS OF THE GROUP OF ANALYTIC SYMPLECTIC DIFFEOMORPHISMS OF A SURFACE

We show that if M is a compact oriented surface of genus 0 and G is a subgroup of Symp(mu)(omega) (M) that has an infinite normal solvable subgroup, then G is virtually abelian. In particular the centralizer of an infinite order f epsilon Symp(mu)(omega) (M) is virtually abelian. Another immediate corollary is that if G is a solvable subgroup of Symp(mu)(omega) (M) then G is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Symp(mu)(omega) (M).