Non-determinism in group actions and topological minimal self-joinings

Abstract: We study the topological and geometrical aspects of non-determinism in group actions, generalizing the notion of non-expansive direction for abelian actions. Under suitable conditions on the acting group, we establish restrictions on the collection of non-deterministic horoballs (or half-spaces) for any action. We illustrate our results in the case of free and nilpotent groups, expanding upon known cases for abelian groups. We then apply these findings to standing problems concerning minimal self-joinings in topological dynamics, which are of independent interest.