Study of the structure of the Markov-Zariski topology of a group and convergence properties of compact-like topological groups

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We prove that Markov and Zariski topologies coincide for abelian groups, and we provide the description of the Markov-Zariski topology of an abelian group. Based on this description, we characterize counatable potentially dense susbets of abelian groups of size at most the continuum, as well as uncountable potentially dense subsets of torsion and divisible abelian groups. We also find a necessary and sufficient condition for the coincidence of Markov and Zariski topologies in the non-commutative case. The theory of group-valued function spaces is developed.