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

TL;DR

This paper characterizes certain classes of precompact abelian groups that lack non-trivial quasi-convex null sequences, providing examples and exploring properties under various minimality and compactness conditions.

Contribution

It offers a comprehensive characterization of groups without non-trivial quasi-convex null sequences across multiple classes, including minimal and pseudocompact groups.

Findings

Characterization of groups with no non-trivial quasi-convex null sequences

Existence of minimal pseudocompact abelian groups with this property

Construction of countably compact minimal abelian groups under Martin's Axiom

Abstract

In this paper, we study precompact abelian groups G that contain no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 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) \omega-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.