  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.
