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On the width of verbal subgroups of certain free constructions. (English. Russian original) Zbl 0941.20017

Algebra Logika 36, No. 5, 494-517 (1997); translation in Algebra Logic 36, No. 5, 288-301 (1997).
The width \(\text{wid}(G,V)\) of the verbal subgroup \(V(G)\) relative to \(V\), defined in a group \(G\) by a set \(V\) of words, is the least \(m\in\mathbb{N}\cup\{+\infty\}\) such that every element of \(V(G)\) is represented as a product of \(\leq m\) values of words in \(V\). The main result of the article is as follows: Assume that, in the HNN-extension \(G^*=\text{gr}(G,t\mid t^{-1}At=B,\;\varphi)\), the subgroups \(A\) and \(B\) are distinct from the base group \(G\); then every subgroup \(V(G^*)\) defined by a finite proper set \(V\) of words has infinite width relative to \(V\).

MSC:

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E10 Quasivarieties and varieties of groups
20F05 Generators, relations, and presentations of groups
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