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^ω_μ (M) that has an infinite normal solvable subgroup, then G is virtually abelian. In particular the centralizer of an infinite order f ∈ Symp^ω_μ(M) is virtually abelian. Another immediate corollary is that if G is a solvable subgroup of Symp^ω_μ(M) then G is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Symp^ω_μ(M).