Ellipticity of words
type: Item
instance of: scholarly article
title: Ellipticity of words
zbMATH Open document ID: 1436.20063
author: Consuelo Martínez
published in: Journal of Algebra
publication date: 2018-02-28T00:00:00Z
full work available at URL: http://hdl.handle.net/10651/43012
review text: Let \(G\) be a pro-\(p\) group and \(w=w(x_1, \ldots ,x_n)\in F_n\) an abstract word on \(n\) letters. It is said that \(w\) is elliptic (or has finite verbal width) on \(G\) if \(\langle \overline{w(G)}\rangle =w(G)^{\pm 1}\stackrel{n)}{\cdots} w(G)^{\pm 1}\) for some \(n\geqslant 1\), where \(w(G)=\{ w(g_1, \ldots ,g_n) \mid g_i\in G\}\subseteq G\) is the set of values of \(w\) in \(G\), and \(\langle \overline{w(G)}\rangle\) is the closure of the subgroup of \(G\) generated by \(w(G)\). Equivalently, \(w\) is elliptic if and only if the discrete subgroup of \(G\) generated by \(w(G)\) is closed. It is known that, on some important classes of groups (like \(p\)-analytic pro-\(p\) groups, the Notingham group, etc), all words are elliptic. The paper under review contributes in this direction proving that all pro-\(p\) completions of finitely generated residually-\(p\) torsion groups also belong to this family: all words are elliptic. Further, the author also proves that in such groups, any multilinear word (see the technical definition in the paper) is strongly elliptic (a somehow stronger condition of ellipticity).
reviewed by: Enric Ventura Capell
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 6845381
zbMATH Keywords: multilinear words, ellipticity, strong ellipticity
MaRDI profile type: Publication
OpenAlex ID: W2588216430
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DOI: 10.1016/J.JALGEBRA.2017.01.046
description: scientific article; zbMATH DE number 6845381