Zariski topology and Markov topology on groups
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 section is dedicated to the abelian case, when the Zariski topology and the Markov topology coincide. Several applications to long standing problems are recalled here (the Markov problem on the connected topologization of abelian groups, the Comfort-Remus-Protasov problem on minimal almost periodic topologization, a problem of Gabriyelyan on the realization of the von Neumann kernel of topological abelian groups). The last part of the paper concerns the groups whose Markov topology is a group topology, which are exactly the groups with a minimum Hausdorff group topology. Special attention is payed to recent results involving permutation groups and groups of homeomorphisms of compact (metric) spaces.
scientific article; zbMATH DE number 6893997
scientific article