Minimally almost periodic group topology on countable torsion Abelian groups

TL;DR

This paper demonstrates that all unbounded countable torsion Abelian groups can be endowed with a complete Hausdorff minimally almost periodic topology, with specific conditions identified for bounded cases.

Contribution

It establishes the existence of MinAP topologies on all unbounded countable torsion Abelian groups and characterizes when bounded groups admit such topologies.

Findings

Unbounded torsion groups admit MinAP topologies.

Bounded torsion groups admit MinAP topologies iff all Ulm-Kaplansky invariants are infinite.

MinAP topologies can be chosen to be complete in certain cases.

Abstract

For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $\tau$ such that $H$ is the von Neumann radical of $(G,\tau)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.