The Markov and Zariski topologies of some linear groups

We study the Zariski topology ZG, the Markov topology MG and the precompact Markov topology PG of an infinite group G, introduced in Dikranjan and Shakhmatov (2007, 2008, 2010) [6–8]. We prove that PG is discrete for a non-abelian divisible solvable group G, concluding that a countable divisible solvable group G is abelian if and only if MG=PG if and only if PG is non-discrete. This answers Dikranjan and Shakhmatov (2010) [8, Question 12.1], . We study in detail the space (G,ZG) 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) [8] in the case of abelian groups.