The Markov and Zariski topologies of some linear groups

We study the Zariski topology Z_G, the Markov topology M_G and the precompact Markov topology P_G of an infinite group G, introduced in byDikranjan and Shakhmatov (2007--2010). We prove that P_G is discrete for a non-abelian divisible solvable group G, concluding that a countable divisible solvable group G is abelian if and only if M_G = P_G if and only if P_G is non-discrete. This answers a question of Dikranjan and Shakhmatov (2010). We study in detail the space (G,Z_G) for two types of linear groups, obtaining a complete description of various topological properties (as dimension, Noetherianity, etc.). This allows us to distinguish, in the case of linear groups, the Zariski topology defined via words (i.e., the verbal topology in terms of Bryant) from the affine topology usually considered in algebraic geometry. We compare the properties of the Zariski topology of these linear groups with the corresponding ones obtained in Dikranjan and Shakhmatov (2010) in the case of abelian groups.