Countably compact groups without non-trivial convergent sequences

TL;DR

This paper constructs a specific countably compact group in ZFC that lacks non-trivial convergent sequences, and demonstrates that the product of two such groups need not be countably compact, addressing longstanding open problems.

Contribution

It provides the first ZFC construction of a countably compact group without non-trivial convergent sequences and shows that countable compactness is not preserved under product.

Findings

Constructed a countably compact subgroup of 2^c without non-trivial convergent sequences

Proved the existence of two countably compact groups whose product is not countably compact

Abstract

We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.