Ultrafilters on G-spaces

For a discrete group G and a discrete G-space X, we identify the Stone-Čech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use G-invariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on ω, the T -points, and study interrelations between these ultrafilters and some classical ultrafilters on ω. By a G-space, we mean a set X endowed with the action G×X → X : (g, x) 7→ gx of a group G. All G-spaces are supposed to be transitive: for any x, y ∈ X, there exists g ∈ G such that gx = y. If X = G and the action is the group multiplication, we say that X is a regular G-space. Several intersting and deep results in combinatorics, topological dynamics and topological algebra, functional analysis were obtained by means of ultrafilters on groups (see [5–7,12,27,28]). The goal of this paper is to systematize some recent and prove some new results concerning ultrafilters on G-spaces, and point out the key open problems. In sections 1,2 and 3, we keep together all necessary definitions of filters, ultrafilters and the Stone-Čech compactification βX of the discrete space X. We extend the action of G on X to the action of βG on βX, characterize the minimal invariant subsets of βX, define the corona X̌ of X and the ultracompanions of subsets of X. 2010 MSC: 05D10, 22A15, 54H20.