Countably compact group topologies on arbitrarily large free Abelian groups

TL;DR

The paper constructs large free Abelian groups with countably compact topologies, answering a longstanding question by showing such groups can be arbitrarily large and lack nontrivial convergent sequences under certain ultrafilter assumptions.

Contribution

It demonstrates the existence of arbitrarily large countably compact Abelian groups with specific topological properties, solving a problem posed in 1992.

Findings

Existence of large countably compact groups under ultrafilter assumptions

Construction of group topologies without nontrivial convergent sequences

Answering a 1992 open question in topological group theory

Abstract

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.