
This article is cited in 2 scientific papers (total in 2 papers)
On the width of verbal subgroups in some classes of groups
I. V. Dobrynina^{a}, D. Z. Kagan^{b} ^{a} Tula State Pedagogical University
^{b} Moscow State University of Railway Communications
Abstract:
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 à 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 nontrivial 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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UDC:
512.54 Received: 20.10.2015
Citation:
I. V. Dobrynina, D. Z. Kagan, “On the width of verbal subgroups in some classes of groups”, Chebyshevskii Sb., 16:4 (2015), 150–163
Citation in format AMSBIB
\Bibitem{DobKag15}
\by I.~V.~Dobrynina, D.~Z.~Kagan
\paper On the width of verbal subgroups in some classes of groups
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 150163
\mathnet{http://mi.mathnet.ru/cheb439}
\elib{https://elibrary.ru/item.asp?id=25006097}
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http://mi.mathnet.ru/eng/cheb439 http://mi.mathnet.ru/eng/cheb/v16/i4/p150
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This publication is cited in the following articles:

D. Z. Kagan, “Nontrivial pseudocharacters on groups with one defining relation and nontrivial centre”, Sb. Math., 208:1 (2017), 75–89

D. Z. Kagan, “Invariantnye funktsii na svobodnykh gruppakh i spetsialnykh HNNrasshireniyakh”, Chebyshevskii sb., 18:1 (2017), 109–122

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