Centralizers and other virtually abelian subgroups of $\Symp^ω_μ(S^2)$

Details Centralizers and other virtually abelian subgroups of $\Symp^ω_μ(S^2)$ Centralizers and other virtually abelian subgroups of $\Symp^ω_μ(S^2)$

2012-04-18

arxiv.org/abs/1204.3961v1

Mathematics

Dynamical Systems

Scientific paper

Suppose $M$ is a compact oriented surface of genus 0 and $\Symp^\omega_\mu(M)$ denotes the group of analytic symplectic diffeomorphisms of $M$. We show that if $G$ is a subgroup of $\Symp^\omega_\mu(M)$ which has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer $\Cent(f)$ of an infinite order $f \in \Symp^\omega_\mu(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of $\Symp^\omega_\mu(M)$ then $G$ is virtually abelian.

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