SSGP topologies on abelian groups of positive finite divisible rank
Volume 244 / 2019
Abstract
For a subset 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.
