New just-infinite pro-\(p\) groups of finite width and subgroups of the Nottingham group. (Q1881127)

The Nottingham group \(\mathcal N\) is the group of automorphisms of the ring of formal power series in \(t\) over the field \(\mathbb{F}_p\) with \(p\) elements, of the form \(t\mapsto t+\sum_{i\geq 1}a_it^{i+1}\). In the paper under review the author constructs a very interesting infinite family of subgroups of \(\mathcal N\) which are just infinite, nonlinear pro-\(p\) groups of finite width. A pro-\(p\) group \(G\) is just infinite if all of its proper quotients are finite; and it is of finite width if there is a uniform bound on the order of its lower central quotients. The graded Lie algebras associated to these groups, with respect to their lower central series, are all isomorphic to \(\mathfrak{sl}_2(\mathbb{F}_p)\otimes t\mathbb{F}_p[t]\); therefore all the lower central quotients have order \(p^3\). This construction answers two questions of \textit{Y.~Barnea} [Commun. Algebra 30, No. 3, 1293-1303 (2002; Zbl 1011.20034)]. To prove that the groups under study are nonlinear, a new description of the centralizers of elements of order \(p\) in \(\mathcal N\) is given, and the concept of Hausdorff dimension is exploited.