Finite groups acting symplectically on $T^2\times S^2$

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 ${\omega }^{\prime }$ on $T^2\times S^2$ and a finite group acting symplectically and effectively on $(T^2\times S^2,{\omega }^{\prime })$ 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 $Symp(T^2\times S^2,\omega )$ is Jordan. We also give upper and lower bounds for the optimal value of the constant $C$ in Jordan's property for $Symp(T^2\times S^2,\omega )$ depending on the cohomology class represented by $\omega$. Our bounds are sharp for a large class of symplectic forms on $T^2\times S^2$.

24 pages; v2: substantial revision; results improved: we give concrete (often sharp) values for the constants in the estimates in the main theorems; v3: title and abstract changed, included corrections and improvements suggested by the referee, added an appendix with a geometric interpretation of the automorphisms of the Heisenberg group; to appear in Trans. AMS

References in corpus (5)

• Groups acting on manifolds: around the Zimmer program
• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields

Cited by in corpus (1)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic