In 1999th G. Baumslag, A. Miasnikov and V. Remeslennikov formulated the following problem:Preliminary future speakers: V. Remeslennikov, Armin Weiss, A. Rybalov.Let L be the language of group theory, G be a group and LG be the language of group theory with a set of constants from the group G. Let the group G = <G, LG> be equationally noetherian over one variable equations (1-equationally noetherian). Does it follow that the group G is equationally noetherian?
An example of a nilpotent group will be presented, which is 1-equationally noetherian but is not equationally noetherian in general. The proof will be based on the new results on a close relation between the concepts of centralizer dimension and noetherenes by equations for groups.