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

John Franks; Michael Handel

doi:10.3934/jmd.2013.7.369

Article Contents

2013, Volume 7, Issue 3: 369-394. Doi: 10.3934/jmd.2013.7.369

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

John Franks^1, and
Michael Handel^2,

1.
Department of Mathematics, Northwestern University, Evanston, IL 60208-2730
2.
Department of Mathematics and Computer Science, Lehman College, Bronx, NY 10468

Received: December 2012

Revised: September 2013

Published: December 2013

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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$^\omega_\mu(M)$.

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Mathematics Subject Classification: Primary: 37C85; Secondary: 37E30, 37C40.

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