Journal of Modern Dynamics (JMD)

Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface

Pages: 369 - 394, Issue 3, September 2013      doi:10.3934/jmd.2013.7.369

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John Franks - Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States (email)
Michael Handel - Department of Mathematics and Computer Science, Lehman College, Bronx, NY 10468, 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.
Mathematics Subject Classification:  Primary: 37C85; Secondary: 37E30, 37C40.

Received: December 2012;      Revised: September 2013;      Available Online: December 2013.