The Coordinate Group of an Affine Space Over a Rigid Metabelian Pro-p-group

Algebraic Sets in a Finitely Generated 2-Step Solvable Rigid Pro-p-Group

“…Our objective is to receive information on algebraic sets in a finitely generated 2-step solvable rigid pro-p-group G (i.e., sets defined by systems of equations in one variable with coefficients in G), similar to one that was obtained in [12,13] for abstract free metabelian groups and wreath products of two free Abelian groups. We will use the definitions and results in [9] concerning algebraic geometry over profinite groups, and in [8,14] dealing directly with 2-step solvable rigid pro-p-groups. For information on profinite groups, we ask the reader to consult [15].…”

“…Consequently, the Zariski topology on G n is Noetherian and an arbitrary closed set in G n is representable as a union of finitely many irreducible components, each of which is an algebraic set. In [14], it was shown that the entire space G n is irreducible. Here we handle the case where n = 1.…”