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This topology was implicitly introduced by A. Markov in 1944 and later explicitly introduced by R. Bryant, under the name verbal topology. In the last ten years a wealth of new papers appeared, where this topology is given the name Zariski topology for its striking similarity with the Zariski topology studied in algebraic geometry. Actually, one of the main direction of research on this topic (pursued by Baumslag, Myasnikov and Remeslennikov) has as principal objective the development of a counterpart of Algebraic Geometry in abstract groups.

This cycle of lectures will be dedicated to another direction, namely the one undertaken by Markov himself towards the solution of one of his problems: the existence of non-discrete Hausdroff group topologies on the infinite groups. To this end one can make use of another topology, introduced again implicitly by Markov. This topology was explicitly introduced by Dikranjan and Shakhmatov under the name Markov topology. In these terms, Markov's problem can be formulated as follows: is the Markov topology of an infinite group always non-discrete. It is easy to see that the Markov topology is finer than the Zariski topology. Markov showed that they coincide for countable groups and asked if this is always the case. The first infinite group with discrete Markov topology was built by Shelah in 1980 under the assumption of the Continuum Hypothesis. Shortly afterwards Ol'shankij gave an example of a countable group with discrete Zariski topology. Finally, some attention will be dedicated to a recently relevant progress in this line obtained by Ol'shankij and his school, in building groups with preassigned properties of the Zariski topology.