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

TL;DR

This paper constructs examples of finite groups acting smoothly but not symplectically on $T^2\times S^2$, studies the Jordan property of symplectomorphism groups, and explores how symplectic forms influence group actions.

Contribution

It demonstrates the existence of finite groups acting smoothly but not symplectically, and establishes the Jordan property for symplectomorphism groups of $T^2\times S^2$ with bounds depending on the symplectic form.

Findings

Finite groups can act smoothly but not symplectically on $T^2\times S^2$.

Symplectomorphism groups of $T^2\times S^2$ are Jordan for any symplectic form.

Bounds for the Jordan constant depend on the cohomology class of the symplectic form.

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 $C$. Csik\'os, Pyber and Szab\'o 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 $T^2\times S^2$.