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The palindromic width of a free product of groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Valery Bardakov  and
Vladimir Tolstykh
Valery Bardakov
Affiliation:
Institute of Mathematics, Siberian Branch Russian Academy of Science, 630090 Novosibirsk, Russia, e-mail: bardakov@math.nsc.ru
Vladimir Tolstykh
Affiliation:
Department of Mathematics, Yeditepe University, 34755 Istanbul, Turkey, e-mail: vtolstykh@yeditepe.edu.tr
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Abstract

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Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

MSC classification

Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 199 - 208
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bardakov, V., Shpilrain, V. and Tolstykh, V., ‘On the palindromic and primitive width of a free group’, J. Algebra 285 (2005), 574–585.CrossRefGoogle Scholar
[2]Bardakov, V. G., ‘On the width of verbal subgroups of some free constructions’, Algebra Logika 36 (1997), 494–517.CrossRefGoogle Scholar
[3]Bergman, G., ‘Generating infinite symmetric groups’, Bull. London Math. Soc. 38 (2006), 429–440.CrossRefGoogle Scholar
[4]Dobrynina, I. V., ‘On the width in free products with amalgamation’, Mat. Zametki 68 (2000), 353–359.Google Scholar
[5]Faiziev, V., ‘A problem of expressibility in some amalgamated products of groups’, J. Aust. Math. Soc. 71 (2001), 105–115.CrossRefGoogle Scholar
[6]Merzlyakov, Yu. I., ‘Algebraic linear groups as full groups of automorphisms and the closure of their verbal subgroups’, Algebra Logika 6 (1967), 83–94.Google Scholar
[7]Rhemtulla, A. H., ‘A problem of bounded expressibility in free products’, Proc. Cambridge Philos. Soc. 64 (1969), 573–584.CrossRefGoogle Scholar