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Mathematische Annalen

, Volume 370, Issue 1–2, pp 331–380 | Cite as

Finite subgroups of Ham and Symp

  • Ignasi Mundet i RieraEmail author
Article

Abstract

Let \((X,\omega )\) be a compact symplectic manifold of dimension 2n and let \({\text {Ham}}(X,\omega )\) be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant C, depending on X but not on \(\omega \), such that any finite subgroup \(G\subset {\text {Ham}}(X,\omega )\) has an abelian subgroup \(A\subseteq G\) satisfying \([G:A]\le C\), and A can be generated by n elements or fewer. If \(b_1(X)=0\) we prove an analogous statement for the entire group of symplectomorphisms of \((X,\omega )\). If \(b_1(X)\ne 0\) we prove the existence of a constant \(C'\) depending only on X such that any finite subgroup \(G\subset {\text {Symp}}(X,\omega )\) has a subgroup \(N\subseteq G\) which is either abelian or 2-step nilpotent and which satisfies \([G:N]\le C'\). These results are deduced from the classification of the finite simple groups, the topological rigidity of hamiltonian loops, and the following theorem, which we prove in this paper. Let E be a complex vector bundle over a compact, connected, smooth and oriented manifold M; suppose that the real rank of E is equal to the dimension of M, and that \(\langle e(E),[M]\rangle \ne 0\), where e(E) is the Euler class of E; then there exists a constant \(C''\) such that, for any prime p and any finite p-group G acting on E by vector bundle automorphisms preserving an almost complex structure on M, there is a subgroup \(G_0\subseteq G\) satisfying \(M^{G_0}\ne \emptyset \) and \([G:G_0]\le C''\).

Mathematics Subject Classification

57S17 53D05 

Notes

Acknowledgements

I wish to thank A. Jaikin, A. Turull and C. Sáez for useful comments. Special thanks to A. Jaikin for providing the proof of Lemma 4.5, which is much shorter and more efficient than the original one. Many thanks finally to the referee for a detailed and very useful report, for a number of corrections and suggestions to improve the paper, and for providing an alternative and more direct proof of Theorem 6.1.

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Departament d’Àlgebra i Geometria, Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain

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