Finite groups acting symplectically on T2× S2
Abstract
For any symplectic form ω on T2× S2 we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on T2× S2 that are trivial in cohomology but which do not admit any effective symplectic action on (T2× S2,ω). We also prove that for any ω there is another symplectic form ω' on T2× S2 and a finite group acting symplectically and effectively on (T2× S2,ω') which does not admit any effective symplectic action on (T2× S2,ω). A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of T2× S2. A group G is Jordan if there exists a constant C such that any finite subgroup of G contains an abelian subgroup whose index in is at most C. Csik\'os, Pyber and Szab\'o proved recently that the diffeomorphism group of T2× S2 is not Jordan. We prove that, in contrast, for any symplectic form ω on T2× S2 the group of symplectomorphisms Symp(T2× S2,ω) is Jordan. We also give upper and lower bounds for the optimal value of the constant C in Jordan's property for Symp(T2× S2,ω) depending on the cohomology class represented by ω. Our bounds are sharp for a large class of symplectic forms on T2× S2.