A complete solution of Markov's problem on connected group topologies

Dikranjan, Dikran; Shakhmatov, Dmitri

doi:10.1016/j.aim.2015.09.006

Cited by 14 publications

(1 citation statement)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The utility of these topologies for the solution of Markov's problem on potential density will be discussed in §1.3. For further applications of these topologies see [14] (for the solution of the fourth Markov problem of connected topologization of abelian groups) and [15] (for the solution of Protasov-Comfort's problem on minimally almost periodic group topologies).…”

Section: Markov's Problems and Two Topologies On Infinite Groupsmentioning

confidence: 99%

Groups with cofinite Zariski topology and potential density

Bonatto¹,

Dikranjan²,

Toller³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Markov's Problems and Two Topologies On Infinite Groupsmentioning

confidence: 99%