Zariski topology and Markov topology on groups

Abstract

Every group G carries an intrinsically defined (by means of solution sets of one-variable equations) topology ZG, named Zariski topology. It is related to another topology MG having as closed sets all unconditionally closed sets of G, named Markov topology, after Markov who implicitly introduced both topologies in dealing with a series of problems related to group topologies. The aim of this survey is to enlighten the utility of these topologies in resolving Markov problems, as well as other challenging problems in the area of topological groups, mainly related to topologization via group topologies with certain properties. We show that these topologies shelter under the same umbrella as distant issues as abelian groups and highly non-abelian ones, as permutation groups and homeomorphism groups.