Algebraic sets in a finitely generated 2-step solvable rigid pro-\(p\)-group. (Q282119)

Let \(G\) be a pro-\(p\) group which contains a normal subgroup \(K\) such that \(A=G/K\) is abelian and torsion-free, and \(K\) is abelian, and torsion-free over the group algebra \(\mathbf Z_pA\) of \(A\) over the \(p\)-adic integers. For instance free metabelian pro-\(p\) groups of rank larger than one satisfy these conditions. In the paper under review, the algebraic sets in such a group are determined -- these are the sets that can be described via a set of equations in one variable with coefficients in \(G\). Each such set is the union of finitely many irreducible components. The irreducible algebraic sets can be the whole of \(G\); singletons; cosets with respect to \(K\); certain infinite sets \(S\), with the property that the natural projection \(S\to A\) is injective.