Pro-\(p\) groups of rank 3 and the question of Iwasawa. (Q1016496)

Let \(p\) be a prime, and for any finitely generated pro-\(p\) group \(G\) let \(d(G)\) denote the minimal number of (topological) generators for \(G\). The aim of this short paper is to solve a special instance of the following classification problem: determine up to isomorphism all finitely generated pro-\(p\) groups \(G\) with the property that \(d(H)=d(G)\) for all open subgroups \(H\) of \(G\). This problem arises naturally from a more general question, attributed to Iwasawa; cf.\ \textit{D.~Dummit} and \textit{J.~Labute} [Invent. Math. 73, 413-418 (1983; Zbl 0546.20021)]. For given \(n\in\mathbb{N}\), what are the finitely generated pro-\(p\) groups \(G\) such that \(d(H)-n=|G:H|(d(G)-n)\) for every open subgroup \(H\) of \(G\)? Indeed, if one restricts attention to the category of \(p\)-adic analytic pro-\(p\) groups, one deals precisely with the pro-\(p\) groups \(G\) such that \(d(H)=n\) is constant as \(H\) runs through all open subgroups of \(G\). The main result of the paper under review is that, for \(p>3\), the pro-\(p\) groups \(G\) with \(d(H)=3\) for all open subgroups \(H\) of \(G\) are up to isomorphism (i) the free Abelian pro-\(p\) group \(G_0=\mathbb{Z}_p^3\) of rank \(3\) and (ii) the \(3\)-dimensional metabelian groups \(G_1(s)\), \(s\in\mathbb{N}\), admitting presentations \[ \langle x,y_1,y_2\mid [y_1,y_2]=1,\;[y_1,x]=y_1^{p^s},\;[y_2,x]=y_2^{p^s}\rangle. \] The proof given can be sketched as follows. The restriction \(p>3\) allows one to deduce that any pro-\(p\) group of the desired type is torsion-free. The author then makes use of a result of \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 506-515 (1987; Zbl 0626.20022)] which shows that groups of the desired type are soluble. Finally, he uses an effective classification of \(3\)-dimensional soluble torsion-free pro-\(p\) groups for \(p>3\), which was given by \textit{J.\ González-Sánchez} and \textit{B.\ Klopsch} [J. Group Theory 12, No. 5, 711-734 (2009; Zbl 1183.20030)].