Finite groups acting symplectically on $T^2\times S^2$
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- by Ignasi Mundet i Riera PDF
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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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Additional Information
- 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
- MR Author ID: 642261
- Email: ignasi.mundet@ub.edu
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4457-4483
- MSC (2010): Primary 57S17, 53D05
- DOI: https://doi.org/10.1090/tran/6978
- MathSciNet review: 3624417