Finite subgroups of Ham and Symp

Let $(X,\omega)$ be a compact symplectic manifold of dimension $2n$ and let $Ham(X,\omega)$ 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 $G\subset Ham(X,\omega)$ has an abelian subgroup $A\subseteq G$ satisfying $[G:A]\leq C$, 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 $b_1(X)\neq 0$ we prove the existence of a constant $C'$ depending only on $X$ such that any finite subgroup $G\subset Symp(X,\omega)$ has a subgroup $N\subseteq G$ which is either abelian or $2$-step nilpotent and which satisfies $[G:N]\leq C'$. 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 $\langle e(E),[M]\rangle\neq 0$, 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 $M^{G_0}\neq\emptyset$ and $[G:G_0]\leq C"$.