New just-infinite pro-\(p\) groups of finite width and subgroups of the Nottingham group.
type: Item
instance of: scholarly article
title: New just-infinite pro-\(p\) groups of finite width and subgroups of the Nottingham group.
zbMATH Open document ID: 1062.20030
published in: Journal of Algebra
publication date: 2004-10-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 2105802
zbMATH Keywords: Nottingham group
zbMATH Keywords: pro-\(p\) groups of finite width
zbMATH Keywords: lower central series
zbMATH Keywords: centralizers
author: Mikhail Ershov
reviewed by: Andrea Caranti
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.08.012
OpenAlex ID: W2084898916
cites work: Q4534141
cites work: Index-subgroups of the Nottingham group.
cites work: Hausdorff dimension, pro-𝑝 groups, and Kac-Moody algebras
cites work: Graded subalgebras of affine Kac-Moody algebras
cites work: Subgroups of the Nottingham group
cites work: Q2702127
cites work: Q4264394
cites work: On just infinite pro-p-groups and arithmetically profinite extensions of local fields
cites work: On linear just infinite pro-\(p\) groups
cites work: Automorphisms of the Nottingham group
cites work: Linear pro-\(p\)-groups of finite width
cites work: Q2702123
cites work: On some \(\Lambda\)-analytic pro-\(p\) groups
cites work: Q3997989
cites work: Compact subgroups of linear algebraic groups
cites work: The number of conjugacy classes in some quotients of the Nottingham group
cites work: Q2702122
cites work: Narrow algebras and groups
cites work: Q3050630
cites work: Q3864611
DOI: 10.1016/J.JALGEBRA.2003.08.012
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description: scientific article; zbMATH DE number 2105802