Finite groups acting symplectically on T^2 x S^2

Abstract

For any symplectic form $ \omega $ on $ T^2\times S^2$ we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on $ T^2\times S^2$ that are trivial in cohomology but which do not admit any effective symplectic action on $ (T^2\times S^2,\omega )$. We also prove that for any $ \omega $ there is another symplectic form $ \omega '$ on $ T^2\times S^2$ and a finite group acting symplectically and effectively on $ (T^2\times S^2,\omega ')$ which does not admit any effective symplectic action on $ (T^2\times S^2,\omega )$. A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of $ T^2\times S^2$. A group $ G$ is Jordan if there exists a constant $ C$ such that any finite subgroup $ \Gamma $ of $ G$ contains an abelian subgroup whose index in $ \Gamma $ is at most $ C$. Csikós, Pyber and Szabó proved recently that the diffeomorphism group of $ T^2\times S^2$ is not Jordan. We prove that, in contrast, for any symplectic form $ \omega $ on $ T^2\times S^2$ the group of symplectomorphisms $ \mathrm {Symp}(T^2\times S^2,\omega )$ is Jordan. We also give upper and lower bounds for the optimal value of the constant $ C$ in Jordan's property for $ \mathrm {Symp}(T^2\times S^2,\omega )$ depending on the cohomology class represented by $ \omega $. Our bounds are sharp for a large class of symplectic forms on $ T^2\times S^2$.