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


Author: Ignasi Mundet i Riera
Journal: Trans. Amer. Math. Soc. 369 (2017), 4457-4483
MSC (2010): Primary 57S17, 53D05
DOI: https://doi.org/10.1090/tran/6978
Published electronically: February 13, 2017
MathSciNet review: 3624417
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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$.


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Ignasi Mundet i Riera
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: ignasi.mundet@ub.edu

DOI: https://doi.org/10.1090/tran/6978
Received by editor(s): July 15, 2015
Received by editor(s) in revised form: February 29, 2016
Published electronically: February 13, 2017
Additional Notes: This work was partially supported by the (Spanish) MEC Project MTM2012-38122-C03-02.
Article copyright: © Copyright 2017 American Mathematical Society