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