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

Abstract

A 2-step solvable 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 torsionfree Abelian, and the subgroup G 2 is also Abelian and is torsion-free as a ℤ p A-module, where ℤ p A is the group algebra of the group A over the ring of p-adic integers. For instance, free metabelian pro-p-groups of rank ≥ 2 are rigid. We give a description of algebraic sets in an arbitrary 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.