Jordan property and almost fixed points

I will talk about the relation between Jordan's property for (subgroups of) diffeomorphism groups and existence of points with big stabilizer.
A group $H$ is Jordan if there exists a constant $C$ such that any finite subgroup $G$ of $H$ has an abelian subgroup $A$ satisfying $[G:A]leq C$. Let $X$ be a smooth manifold and let $H$ be a subgroup of $Diff(X)$. The pair $(X,H)$ has the almost fixed point property if there is a constant $C$ such that for any finite subgroup $G$ of $H$ there exists a point $xin X$ whose stabilizer $G_x$ satisfies $[G:G_x]leq C$.
Theorem. If $X$ is a compact manifold, possibly with boundary, and the cohomology of $X$ is torsion free and concentrated in even degrees, then $(X,Diff(X))$ has the almost fixed point property.
I will explain how the theorem follows from the fact that $Diff(X)$ is Jordan. Using a result of Petrie and Randall, the theorem implies:
Corollary. Let $Z$ be a real affine manifold, not necessarily compact, and let $Aut(Z)$ denote the group of algebraic automorphisms of $Z$. If the cohomology of $Z$ is torsion free and concentrated in even degrees, then $(Z,Aut(Z))$ has the almost fixed point property.