Rigid Metabelian Pro-p-Groups

A metabelian pro-p-group G is rigid if it has a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian and C = G2 is torsion-free as a ZpA-module. If G is a non-Abelian group, then the subgroup G2, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-p-group is rigid if it is torsion-free, and as G2 we can take either the trivial subgroup or the entire group. We prove that all rigid 2-step solvable pro-p-groups are mutually universally equivalent. Rigid metabelian pro-p-groups can be treated as 2-graded groups with possible gradings (1, 1), (1, 0), and (0, 1). If a group is 2-step solvable, then its grading is (1, 1). For an Abelian group, there are two options: namely, grading (1, 0), if G2 = 1, and grading (0, 1) if G2 = G. A morphism between 2-graded rigid pro-p-groups is a homomorphism φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varphi $$\end{document} : G → H such that Giφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varphi $$\end{document} ≤ Hi. It is shown that in the category of 2-graded rigid pro-p-groups, a coproduct operation exists, and we establish its properties.