Compact-like abelian groups without non-trivial quasi-convex null sequences

In this paper, we study precompact abelian groups G that contain no sequence such that {0}∪{±xn∣n∈N} is infinite and quasi-convex in G, and xn⟶0. We characterize groups with this property in the following classes of groups: (a)bounded precompact abelian groups;(b)minimal abelian groups;(c)totally minimal abelian groups;(d)ω-bounded abelian groups. We also provide examples of minimal abelian groups with this property, and show that there exists a minimal pseudocompact abelian group with the same property; furthermore, under Martin’s Axiom, the group may be chosen to be countably compact minimal abelian.