Jaikin-Zapirain, Andrei On the verbal width of finitely generated pro-\(p\) groups. (English) Zbl 1158.20012 Rev. Mat. Iberoam. 24, No. 2, 617-630 (2008). Let \(w\) be an element of a free group \(F\) on \(k\) independent generators, i.e., a word in \(k\) variables. A \(w\)-value in a group \(G\) is an element of the form \(g=w(g_1,\dots,g_k)^{\pm 1}\), for suitable \(g_1,\dots,g_k\in G\). Let \(w(G)\) be the subgroup of \(G\) generated by the set \(G^{\{w\}}\) of \(w\)-values in \(G\). It is well-known that if \(G\) is profinite, then the (abstract) subgroup \(w(G)\) is closed if and only if the (verbal) width of \(w\) in \(G\) is finite, i.e., there is \(l\) such that every element of \(w(G)\) is the product of at most \(l\) elements of \(G^{\{w\}}\). It is natural to ask which words have finite width in a finitely generated profinite group. The most important contribution in this context is due to N. Nikolov and D. Segal [Ann. Math. (2) 165, No. 1, 239-273 (2007; Zbl 1126.20018)]; they show among others that if \(w\) is a simple commutator, then \(w\) has finite width in any finitely generated profinite group \(G\). In the main result of the paper under review, the author shows if \(w\in F\) is non-trivial, then \(w(H)\) is closed for every finitely generated pro-\(p\) group \(H\) if and only if \(w\notin (F')^pF''\). Moreover, it is shown that every word has finite verbal width in a compact \(p\)-adic group. Reviewer: A. Caranti (Trento) Cited in 18 Documents MSC: 20E18 Limits, profinite groups 22E35 Analysis on \(p\)-adic Lie groups 20F05 Generators, relations, and presentations of groups Keywords:pro-\(p\) groups; verbal subgroups; verbal widths; \(p\)-adic analytic groups; finitely generated profinite groups Citations:Zbl 1126.20018 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid EuDML References: [1] Barnea, Y. and Shalev, A.: Hausdorff dimension, pro-\(p\) groups, and Kac-Moody algebras. 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