Jordan and almost fixed point properties for topological manifolds

I will explain recent results on the Jordan property for homeomorphism groups that generalize most of the presently known results about Jordan diffeomorphism groups. A crucial ingredient in these results is a recent theorem of Csikós, Pyber and Szabó. I will also talk about the following application. Let X be a compact topological manifold, possibly with boundary, with nonzero Euler characteristic. Then there exists a constant $C$ such that for any continuous action of any finite group $G$ on $X$ there is a point in $X$ whose stabilizer has index in $G$ not bigger than $C$.