Finite subgroups of Ham and Symp

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 $[G:G_0]\le C''$.