Finite groups acting symplectically on 𝑇²×𝑆²

Abstract

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