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

Abstract

Let $w=w(x_1,\dots ,x_r)$ be a lower central word or a derived word. We show that the word $w(u_1,\dots ,u_r)$ is concise whenever $u_1,\dots ,u_r$ 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 $w(x_1^{n_1},\dots ,x_r^{n_r})$, 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 $N_1,\dots ,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\dots ,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e. $w(N_1,\dots ,N_r)$, is finite.