Dynamical equivalences on G∗

For every infinite discrete group G, the remainder G∗=βG∖G of the Stone–Čech compactification βG of G has a natural structure of G-space. The orbit equivalence E on G∗ ((x,y)∈E⇔gx=y for some g∈G) produces the following three derived equivalences on G∗(), where Ex, Ey are E-equivalence classes containing x and y, is the smallest by inclusion equivalence on G∗ containing E such that every -equivalence class is closed, is the smallest by inclusion closed in G∗×G∗ equivalence on G∗ containing E, and the relation, y∈clEz, which is an equivalence if G is countable.We study the interrelations between the classes of these equivalences and the principal left ideals of the semigroup βG.