Zariski topology and Markov topology on groups
type: Item
instance of: scholarly article
title: Zariski topology and Markov topology on groups
zbMATH Open document ID: 1478.22002
author: Dikran Dikranjan
author: D. Toller
published in: Topology and its Applications
publication date: 2018-06-25T00:00:00Z
review text: This is a second survey of the same authors on the Zariski topology and the Markov topology. Indeed, [\textit{D. Dikranjan} and \textit{D. Toller}, in: Ischia group theory 2010. Proceedings of the conference in group theory, Ischia, Naples, Italy, April 14--17, 2010. Hackensack, NJ: World Scientific. 87--130 (2012; Zbl 1295.22002)] contains the progress obtained in this field within 2010, while the present paper concerns recent progress with special attention to the non-abelian case. For a group \(G\), the Markov topology \(\mathfrak M_G\) is the infimum of all Hausdorff group topologies on \(G\) taken in the lattice of all topologies on \(G\). This topology was introduced by Markov to address the problem of the existence of an infinite non-topologizable group (i.e., an infinite group such that the only Hausdorff group topology on it is the discrete one). The Zariski topology \(\mathfrak Z_G\) on a group \(G\) is an intrinsically defined topology, which is related to the Markov topology. In particular, one of the problems by Markov can be stated as: is the equality \(\mathfrak Z_G=\mathfrak M_G\) always true? This survey explains the utility of the Zariski topology and the Markov topology in solving open problems due to Markov and other problems in the area of topological groups concerning topologization of groups via group topologies with certain properties. First, the authors recall two of the problems by Markov that triggered these two topologies, and the connection of the topologies with the problems. They further recall some partial Zariski topologies, such as the centralizer topology and the monomial topology. Then they discuss the behavior of the Zariski topology and the Markov topology under taking subgroups, quotients and products. Furthermore, they consider stability properties of the class of groups whose Zariski topology is Noetherian (recall that a topological space is Noetherian if it satisfies the ascending chain condition on open sets). An entire sec…
reviewed by: Anna Giordano Bruno
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F70
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 6893997
zbMATH Keywords: topologizable group
zbMATH Keywords: non-topologizable group
zbMATH Keywords: Noetherian space
zbMATH Keywords: Zariski topology
zbMATH Keywords: Markov topology
zbMATH Keywords: centralizer topology
zbMATH Keywords: minimally almost periodic group
zbMATH Keywords: von Neumann kernel
MaRDI profile type: Publication
full work available at URL: j.topol.2018.03.025
OpenAlex ID: W2795297530
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Zariski topology and Markov topology on groups (Wikidata QID)
Wikidata QID: Q130060972

Zariski topology and Markov topology on groups (DOI)
DOI: 10.1016/J.TOPOL.2018.03.025

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