SSGP topologies on abelian groups of positive finite divisible rank
Volume 244 / 2019
Abstract
For a subset $A$ of a group $G$, we denote by $\def\grp#1{\langle{#1}\rangle}\grp{A}$ the smallest subgroup of $G$ containing $A$ and let $\def\Cyc{\mathrm{Cyc}}\Cyc(A)=\{x\in G: \def\grp#1{\langle{#1}\rangle}\grp{\{x\}}\subseteq A\}$. A topological group $G$ is SSGP if $\def\grp#1{\langle{#1}\rangle}\grp{\def\Cyc{\mathrm{Cyc}}\Cyc(U)}$ is dense in $G$ for every neighbourhood $U$ of the identity of $G$. The SSGP groups form a proper subclass of the class of minimally almost periodic groups.
Comfort and Gould asked about a characterization of abelian groups which admit an SSGP group topology. An “almost complete” characterization was found by Dikranjan and the first author. The remaining case is resolved here. As a corollary, we give a positive answer to another question of Comfort and Gould by showing that if an abelian group admits an SSGP($n$) group topology for some positive integer $n$, then it admits an SSGP group topology as well.
