Countable dense homogeneity and topological groups

TL;DR

This paper constructs a new example of a non-Polish topological group that is countable dense homogeneous, using ZFC set theory, and explores related conjectures about the structure of such groups.

Contribution

It provides the first ZFC example of a non-Polish countable dense homogeneous topological group and investigates its properties and related conjectures.

Findings

Constructed a dense subgroup of aZ^f of size  that is a bb-set.

Proved the subgroup is a bb-set and dense in aZ^f.

Formulated and proved a special case of a conjecture about Baire topological groups.

Abstract

Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^\omega$ of size $\mathfrak{b}$ that is a $\lambda$-set. We also conjecture that every countable dense homogenous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.