Partially Divisible Completions of Rigid Metabelian Pro-p-groups

Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-p-group G is said to be rigid if it contains a normal series of the form G = G 1 ≥ G 2 ≥ G 3 = 1 such that the factor group A = G/G 2 is torsion-free Abelian, and G 2 being a Z p A-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-p-groups. In so doing, we need to exit the class of pro-p-groups, since even the completion of a torsion-free nontrivial Abelian pro-p-group is not a pro-p-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group A. Indeed, A is simply structured: it is isomorphic to a direct sum of copies of Z p . A second factor, i.e., the group G 2 , is completed to a vector space over a field of fractions of a ring Z p A, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.