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Transactions of the American Mathematical Society

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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
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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  • [1] Miguel Abreu and Dusa McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000), no. 4, 971-1009 (electronic). MR 1775741,
  • [2] Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. MR 2273508
  • [3] E. Breuillard, An exposition of Jordan's original proof of his theorem on finite subgroups of $ \mathrm {GL}_n(\mathbb{C})$, preprint available at˜breuilla/Jordan.pdf.
  • [4] Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077
  • [5] Weimin Chen, On the orders of periodic diffeomorphisms of 4-manifolds, Duke Math. J. 156 (2011), no. 2, 273-310. MR 2769218,
  • [6] Weimin Chen and Slawomir Kwasik, Symplectic symmetries of 4-manifolds, Topology 46 (2007), no. 2, 103-128. MR 2313067,
  • [7] Weimin Chen and Slawomir Kwasik, Symmetric symplectic homotopy $ K3$ surfaces, J. Topol. 4 (2011), no. 2, 406-430. MR 2805997,
  • [8] B. Csikós, L. Pyber, E. Szabó, Diffeomorphism groups of compact $ 4$-manifolds are not always Jordan, preprint arXiv:1411.7524.
  • [9] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1962 original. MR 2215618
  • [10] Iku Nakamura, McKay correspondence, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 267-298. MR 2500567
  • [11] Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, pp. 29-55 (French). MR 0042768
  • [12] H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765
  • [13] David Fisher, Groups acting on manifolds: around the Zimmer program, Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 72-157. MR 2807830,
  • [14] Richard M. Hain, The geometry of the mixed Hodge structure on the fundamental group, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 247-282. MR 927984
  • [15] Helmut Hofer, Véronique Lizan, and Jean-Claude Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997), no. 1, 149-159. MR 1630789,
  • [16] C. Jordan, Mémoire sur les équations différentielles linéaires à intégrale algébrique, J. Reine Angew. Math. 84 (1878) 89-215.
  • [17] François Lalonde and Dusa McDuff, The classification of ruled symplectic $ 4$-manifolds, Math. Res. Lett. 3 (1996), no. 6, 769-778. MR 1426534,
  • [18] François Lalonde and Dusa McDuff, $ J$-curves and the classification of rational and ruled symplectic $ 4$-manifolds, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 3-42. MR 1432456
  • [19] Dusa McDuff, The structure of rational and ruled symplectic $ 4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679-712. MR 1049697,
  • [20] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373431
  • [21] Ignasi Mundet i Riera, Jordan's theorem for the diffeomorphism group of some manifolds, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2253-2262. MR 2596066,
  • [22] Ignasi Mundet i Riera, Finite group actions on 4-manifolds with nonzero Euler characteristic, Math. Z. 282 (2016), no. 1-2, 25-42. MR 3448372,
  • [23] I. Mundet i Riera, Finite group actions on homology spheres and manifolds with nonzero Euler characteristic, preprint arXiv:1403.0383.
  • [24] I. Mundet i Riera, Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds, to appear in Transformation Groups, preprint arXiv:1412.6964.
  • [25] D. V. Osipov, The discrete Heisenberg group and its automorphism group, Mat. Zametki 98 (2015), no. 1, 152-155 (Russian); English transl., Math. Notes 98 (2015), no. 1-2, 185-188. MR 3399166,
  • [26] D.V. Osipov, A.N. Parshin, Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields, preprint arXiv:1510.02423.
  • [27] Vladimir L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, Affine algebraic geometry, CRM Proc. Lecture Notes, vol. 54, Amer. Math. Soc., Providence, RI, 2011, pp. 289-311. MR 2768646
  • [28] Vladimir L. Popov, Jordan groups and automorphism groups of algebraic varieties, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., vol. 79, Springer, Cham, 2014, pp. 185-213. MR 3229352,
  • [29] Yuri G. Zarhin, Theta groups and products of abelian and rational varieties, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 299-304. MR 3165026,

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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

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

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