An elementary algebraic set of a group G is a soluton of the equation

x^{e_1} a_1 x^{e_2} a_2 ... a_{n-1} x^{e_n} = a_n,

where a_1,...,a_n are elements of G and e_1,...,e_n are either +1 or -1. 
The intersection of finite unions of elementary algebraic  sets is the 
collection of closed subsets of the unique topology on G called Zariski 
topology. 

A subset of a group G is called absolutely closed if it is closed in *every*
Hausdorff group topology on G. The collection of absolutely closed sets coincides
with the family of closed sets of the unique topology on G called Markov topology. 

It is a major open problem of Markov, going back more then 60 years, to determine
whether these two topologies coincide. We present a survey of recent developments
obtained by the speaker in a joint work with Dikran Dikranjan (Univ. of Udine, Italy).