Locally compact abelian groups admitting non-trivial quasi-convex null sequences

TL;DR

This paper characterizes locally compact abelian groups that do not admit non-trivial quasi-convex null sequences, showing they are essentially built from open subgroups of 2- or 3-torsion elements.

Contribution

It provides a complete characterization of such groups, linking the absence of non-trivial quasi-convex null sequences to their subgroup structure.

Findings

Groups with no non-trivial quasi-convex null sequences have specific subgroup structures.

Such groups contain open compact subgroups isomorphic to Z_2^kappa or Z_3^kappa.

The characterization involves conditions on 2- and 3-torsion subgroups.

Abstract

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains 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; (ii) one of the subgroups {g \in G : 2g=0 and {g \in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^\kappa or Z_3^\kappa for some cardinal \kappa.