Universal properties of group actions on locally compact spaces

We study universal properties of locally compact G-spaces for countable infinite groups G. In particular we consider open invariant subsets of the G-space βG, and their minimal closed invariant subspaces. These are locally compact free G-spaces, and the latter are also minimal. We examine the properties of these G-spaces with emphasis on their universal properties.As an example of our results, we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non-compact Cantor set.