Conciseness on normal subgroups and new concise words from lower central and derived words

Document Type : Dedicated to Prof. D. J. S. Robinson


Department of Mathematics,‎ University of the Basque Country UPV/EHU, Bilbao‎, ‎Spain.


Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables, thus proving a generalized version of a conjecture of Azevedo and Shumyatsky. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e.\ $w(N_1,\ldots,N_r)$, is finite.


Main Subjects

[1] A. Abdollahi and F. G. Russo, On a problem of P. Hall for Engel words, Arch. Math. 97 (2011), no. 5, 407–412.
[2] J. Azevedo and P. Shumyatsky, On finiteness of verbal subgroups, J. Pure Appl. Algebra, 226 (2022), no. 6, Paper No. 106976, 11 pages.
[3] R. Baer, Endlichkeitskriterien für Kommutatorgruppen, Math. Ann. 124 (1952) 161--177.
[4] C. Delizia, P. Shumyatsky, A. Tortora and M. Tota, On conciseness of some commutator words, Arch. Math. (Basel) 112 (2019), no. 1, 27--32.
[5] E. Detomi, M. Morigi and P. Shumyatsky, Words of Engel type are concise in residually finite groups, Bull. Math. Sci. 09 (2019), no. 2, 1950012, 19 pages.
[6] G.A. Fernández-Alcober and M. Morigi, Outer commutator words are uniformly concise, J. London Math. Soc. (2)
82 (2010), no. 3, 581--595.
[7] G. A. Fernández-Alcober, M. Morigi and G. Traustason, A note on conciseness of Engel words, Comm. Algebra 40 (2012), no. 7, 2570–2576.
[8] G. A. Fernández-Alcober and P. Shumyatsky, On bounded conciseness of words in residually finite groups, J. Algebra 500 (2018) 19--29.
[9] I. de las Heras and M. Morigi, Lower central words in finite p--groups, Publ. Mat. 65 (2021), no. 1, 243–269.
[10] S. V. Ivanov, P. Hall’s conjecture on the finiteness of verbal subgroups, Izv. Vyssh. Uchebn. Zaved. 325 (1989), no. 6, 60--70.
[11] A. Yu. Olshanskii, Geometry of defining relations in groups, Kluwer Academic Publishers Group, Dordrecht, 1991.
[12] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer-Verlag, New York--Berlin, 1972.
[13] D. Segal, Words: notes on verbal width in groups, Cambridge University Press, Cambridge, 2009.
[14] R. F. Turner--Smith, Finiteness conditions for verbal subgroups, J. London Math. Soc. 41 (1966) 166–176.
[15] J. Wilson, On outer--commutator words, Canad. J. Math. 26 (1974) 608--620.