Mild 2-relator pro-\(p\)-groups. (Q554251)

Let \(G\) be a finitely presented pro-\(p\) group and assume that the cup product \[ H^1(G,\mathbb Z/p\mathbb Z)\otimes H^1(G,\mathbb Z/p\mathbb Z)\overset{\cup}\twoheadrightarrow H^2(G,\mathbb Z/p\mathbb Z)\tag{1} \] is surjective. It is known that its cohomological dimension is \(2\) and good computations of its Lie algebra are available whenever \(G\) is \textit{mild}, a condition which involves the holonomy algebra \(B=B(G)\) [see \textit{J. Labute}, J. Reine Angew. Math. 596, 155-182 (2006; Zbl 1122.11076)]. \(B\) is an \(\mathbb F_p\)-algebra with \(d=\dim_{\mathbb F_p}H^1(G,\mathbb Z/p\mathbb Z)\) generators and \(r=\dim_{\mathbb F_p}H^2(G,\mathbb Z/p\mathbb Z)\) relations. The aim of the paper under review is to study in detail which pro-\(2\) groups with \(d=4\) and \(r=2\) satisfying (1) are mild (the case \(p\neq 2\) is notably easier, [see \textit{D. J. Anick}, J. Algebra 78, 120-140 (1982; Zbl 0502.16002)]). The authors address the problem via an explicit description of all possible (isomorphism classes of) \(\mathbb F_p\)-algebras which occur as holonomy algebra of pro-\(2\)-groups with \(d=4\) generators, \(r=2\) relations satisfying (1). This they do through Proposition 1 and, using MAGMA, they write down a full list of such algebras. Combining criteria of Anick [loc. cit.] and of \textit{P. Forré} [J. Reine Angew. Math. 658, 173-192 (2011; Zbl 1292.12004)], they can eventually check which element of the list -- consisting of \(59\) entries -- is mild (see Theorem 4). The main application of the above explicit results is to algebraic properties of Galois groups of number fields. Let \(k\) be a totally imaginary number field, \(S\) be a finite set of places of \(k\) and \(G_S(2)\) be the Galois group over \(k\) of the maximal \(2\)-extension of \(k\) (inside a fixed algebraic closure) unramified outside of \(S\). When \(S\supseteq\{\mathfrak p\text{ such that }\mathfrak p\mid 2\}\), it has been known since a long time that \(\mathrm{cd}_2(G_S(2))\leq 2\) [see, for instance, the book of \textit{J. Neukirch, A. Schmidt} and \textit{K. Wingberg}, Cohomology of number fields. 2nd ed. Grundlehren der Mathematischen Wissenschaften 323. Berlin: Springer (2008; Zbl 1136.11001)]. Only recently, however, \textit{A. Schmidt} has been able to prove quite precise results if \(S\not\supseteq\{\mathfrak p\text{ such that }\mathfrak p\mid 2\}\) [see J. Reine Angew. Math. 640, 203-235 (2010; Zbl 1193.14041)]. The authors exploit their explicit list to address the question of mildness for such Galois groups. As an application of their theoretic results they can exhibit, mainly using standard class field theory, that:{\parindent=6mm \begin{itemize}\item[---] If \(k=\mathbb Q(\sqrt{-7})\) and \(S=\{\mathfrak p,\overline{\mathfrak p},\mathfrak q\}\) where \(\mathfrak p,\overline{\mathfrak p}\) are the primes above \(2\) and \(\mathfrak q\) is one of the primes above \(11\), then \(G_S(2)\) is mild;\item[---] If \(k=\mathbb Q(\sqrt{-7})\) and \(S=\{\mathfrak p,\overline{\mathfrak p},\mathfrak q\}\) where \(\mathfrak p,\overline{\mathfrak p}\) are the primes above \(2\) and \(\mathfrak q\) is the unique prime above \(3\), then \(G_S(2)\) is mild;\item[---] If \(k=\mathbb{Q}(\root 3\of 3)\) and \(S=\{\mathfrak p,\overline{\mathfrak p},\mathfrak q\}\) where \(\mathfrak p,\overline{\mathfrak p}\) are the primes above \(2\) and \(\mathfrak q\) is the real prime of \(k\), then \(G_S(2)\) cannot be mild, as \(k\) is not totally imaginary; hence, it contains an element of order \(2\) and its cohomological dimension is infinite. Quite remarkably, though, the authors show that \(G_S(2)\) corresponds to the 54-th element in their list (indeed, a non-mild entry!). \end{itemize}} The paper falls in six sections followed by an appendix. The first section is introductory, Sections 2 and 3 contain a theoretic description of the general shape a holonomy algebra can take and Sections 4 and 5 are concerned with applications of Anick's and Forré's criteria to check for mildness. Section 6 contains the arithmetic applications and the Appendix presents the MAGMA computations of all possible holonomy algebras for \(p=2\), \(d=4\), \(r=2\) for groups fulfilling (1) following the description of Section 3.