Geodesic
On the width of verbal subgroups in some classes of groups
Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 150-163

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In this paper the problem of the width for verbal subgroups in different classes of groups is considered. We give a review the results obtained in this direction. The width of the verbal subgroups $V (G) $ is equal to а least value of $m\in \mathcal{N}\bigcup \{+\infty \}$ such that every element of the subgroup $ V (G) $ is represented as the product of at most $m$ values of words $V^{\pm 1}.$ The results about the width of verbal subgroups for free products and other free group constructions, such as free products with amalgamation and $HNN$-extensions are indicated. A. H. Rhemtulla solved the question of conditions for infinity of the width of any proper verbal subgroups in free products. V. G. Bardakov and I. V. Dobrynina received similar results for the free products with amalgamation and $HNN$-extensions, for which associated subgroups are different from the base group. Also, V. G. Bardakov completely solved the problem of the width of verbal subgroups in the group of braid. Many mathematicians studied the width of verbal subgroups generated by words from commutator subgroup for some classes of groups. R. I. Grigorchuk found conditions for infinity such verbal subgroups of free products with amalgamation and $HNN$-extensions, for which associated subgroups are \linebreak different from the base group. D. Z. Kagan obtained the corresponding results on width of verbal subgroups generated by words from commutator subgroup for groups with one defining relation and two generators, having a non-trivial center. Authors obtained the results about infinity of the width of verbal subgroups for groups with certain presentations, as well as for anomalous products of various types of groups. Also many results about verbal subgroups of Artin and Coxeter groups and graph groups are considered in the article. Bibliography: 25 titles.
Keywords: width of verbal subgroup, amalgamated free products, $HNN$-extensions. 
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     author = {I. V. Dobrynina and D. Z. Kagan},
     title = {On the width of verbal subgroups in some classes of groups},
     journal = {\v{C}eby\v{s}evskij sbornik},
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I. V. Dobrynina; D. Z. Kagan. On the width of verbal subgroups in some classes of groups. Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 150-163. http://geodesic.mathdoc.fr/item/CHEB_2015_16_4_a7/