Topological dynamics beyond Polish groups

When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be robust dividing line in complexity topological dynamics $G$. We introduce class groups, CAP which provides neat generalization this all groups. prove number characterizations class, having very different flavors, and use these that groups enjoys nice closure properties. As concrete application, we compute homeomorphism several scattered spaces, building on recent work Gheysens.