Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of $S^2$

We show that if $M$ is a compact oriented surface of genus 0 and $G$ is a subgroup of $\Symp^ω_μ(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in \Symp^ω_μ(M)$ 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 $\Symp^ω_μ(S^2).$

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