If \((X,\omega)\) is a symplectic manifold, let \(\mathrm{Ham}(X,\omega)\) be the group of Hamiltonian diffeomorphisms and \(\mathrm{Symp}(X,\omega)\) the group of symplectic diffeomorphisms of \((X,\omega)\). \(\mathrm{Symp}_0(X,\omega)\subseteq\mathrm{Symp}(X,\omega)\) and \(\mathrm{Diff}_0(X)\subseteq\mathrm{Diff}(X)\) are the identity components. If \(C\) and \(d\) are natural numbers, then a group \({\mathcal G}\) is said to be \((C,d)\)-Jordan if each finite subgroup \(G\subseteq{\mathcal G}\) has an abelian subgroup \(A\subseteq G\) satisfying \([G : A]\leq C\) and such that \(A\) can be generated by \(d\), or fewer elements.
In this paper, the author is interested in the finite subgroups of \(\mathrm{Ham}(X,\omega)\) or \(\mathrm{Symp}(X,\omega)\) for an arbitrary compact symplectic manifold \(X\). It is shown that if \((X,\omega)\) is a \(2n\)-dimensional compact and connected symplectic manifold, then \(\mathrm{Ham}(X,\omega)\) is \((C,n)\)-Jordan for some \(C\) depending only on \(H^*(X)\), where \(H^*(X)\) denotes the integral cohomology of \(X\). This result implies that if \(\dim X=2n\) and the flux homomorphism \(\pi_1(\mathrm{Symp}_0(X,\omega))\to H^1(X;\mathbb R)\) vanishes, then \(\mathrm{Symp}_0(X,\omega)\) is \((C,n)\)-Jordan for some \(C\) depending only on \(H^*(X;\mathbb Z)\). Also, the author proves that if \((X,\omega)\) is a \(2n\)-dimensional compact and connected symplectic manifold satisfying \(b_1(X)= 0\), then \(\text{Symp}(X,\omega)\) is \((C,n)\)-Jordan for some \(C\) which only depends on \(H^*(X)\). The stronger result states that if \((X,J)\) is a \(2n\)-dimensional almost complex, compact, and connected smooth manifold satisfying \(b_1(X)= 0\), and there exists \(\omega\in H^2(X;\mathbb R)\) such that \(\omega^n\neq 0\), then the group \(\mathrm{Diff}(X,J)\) of all diffeomorphisms preserving the almost complex structure \(J\) on \(X\) is \((C,n)\)-Jordan for a constant \(C\) depending only on \(H^*(X)\).
It is not known whether the symplectomorphism group of every compact symplectic manifold is Jordan. The author proves the weaker result stating that if \((X,\omega)\) is a compact and connected symplectic manifold, then there exists a constant \(C\) depending only on \(H^*(X)\) with the property that any finite subgroup \(\Gamma\subset\mathrm{Symp}(X,\omega)\) has a subgroup \(N\subseteq\Gamma\) satisfying \([\Gamma : N]\leq C\) and \(N\) is either abelian or \(2\)-step nilpotent. The above results are consequences of the following property: If \(E\) is a complex vector bundle over a compact, connected, smooth, and oriented manifold \(M\), the real rank of \(E\) is equal to the dimension of \(M\), and \(\langle e(E),[M]\rangle\neq 0\), 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 \(J\) on \(M\), there is a subgroup \(G_0\subseteq G\) satisfying \(M^{G_0}\neq\varnothing\) and \([G : G_0]\leq C\).