Groups with cofinite Zariski topology and potential density

TL;DR

This paper explores groups with cofinite Zariski topology, extending previous abelian group results to non-abelian groups, and proposes a conjecture that such groups are either abelian or finite.

Contribution

It generalizes Tkachenko-Yaschenko's characterization to non-abelian groups and introduces a partial Zariski topology and stronger torsion-free conditions.

Findings

The Tkachenko-Yaschenko theorem does not hold for non-abelian groups.

Strong restrictions are found on non-abelian groups with cofinite Zariski topology.

A conjecture is proposed that such groups are either abelian or finite.

Abstract

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is either almost torsion-free or of prime exponent). The authors connected this fact to Markov's notion of potential density and the existence of pairs of independent group topologies. Inspired by their work, we examine the class C of groups having cofinite Zariski topology in the general case, obtaining a number of very strong restrictions on these groups in the non-abelian case which suggest the bold conjecture that a group with cofinite Zariski topology is necessarily either abelian or finite. We show that Tkachenko-Yaschenko theorem fails in the non-abelian case and we offer a natural counterpart in the general case using a partial Zariski topology and an appropriate stronger version of the property almost torsion-free.