Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface
John Franks - Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States (email) Abstract: We show that if $M$ is a compact oriented surface of genus $0$ and $G$ is a subgroup of Symp$^\omega_\mu(M)$ that has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer 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. We also prove a special case of the Tits Alternative for subgroups of Symp$^\omega_\mu(M)$.
Keywords: Surface diffeomorphism groups, area-preserving, entropy.
Received: December 2012; Revised: September 2013; Available Online: December 2013. |