Densities and Weights of Quotients of Precompact Abelian Groups

TL;DR

This paper investigates the sizes of quotient groups of precompact abelian groups, establishing bounds on their densities and weights, and constructs examples with prescribed quotient group weights, addressing a longstanding open problem.

Contribution

It determines the minimal cardinalities for quotients of precompact abelian groups and characterizes all possible weights of their infinite proper quotients.

Findings

If $2^{<\mathfrak{c}}=\mathfrak{c}$, then the minimal density is $\mathfrak{c}$ and the minimal weight is $2^{\mathfrak{c}}$.

For any subset of $[\omega, \mathfrak{c}]$, there exists a precompact abelian group with that set as its quotient weights.

A non-totally disconnected locally compact group may lack separable quotient groups, answering an open problem.

Abstract

The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient groups. However, as dense subgroups of compact groups, precompact groups may admit no non-trivial metrizable quotient groups, so also no non-trivial separable quotient groups. In this paper, we study the least cardinal $\mathfrak{m}$ (resp. $\mathfrak{n}$) such that every infinite precompact abelian group admits a quotient group with density character $\leq \mathfrak{m}$ (resp. with weight $\leq \mathfrak{n}$). It is shown that if $2^{<\mathfrak{c}}=\mathfrak{c}$, then $\mathfrak{m}=\mathfrak{c}$ and $\mathfrak{n}=2^\mathfrak{c}$. A more general problem is to describe the set $QW(G)$ of all possible weights of infinite proper quotient groups of a precompact abelian group $G$. We prove that for every subset $E$ of the interval $[\omega, \mathfrak{c}]$, there exists a precompact abelian group $G$ with $QW(G)=E$. If $\omega\in E$, then $G$ can be chosen to be pseudocompact. In an appendix, we give an example to show that a non-totally disconnected locally compact group may admit no separable quotient groups. This answers an open problem posed in \cite{LMT}.