Conciseness on normal subgroups and new concise words from outer commutator words

Let w=w(x₁,…,xᵣ) be an outer commutator word. We show that the word w(u₁,…,uᵣ) is concise whenever u₁,…,uᵣ are non-commutator words in disjoint sets of variables. This applies in particular to words of the form w(x₁n₁,…,xᵣnᵣ), where the nᵢ 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 N₁,…,Nᵣ are normal subgroups of a group G and the set of all values w(g₁,…,gᵣ) with gᵢ∈ Nᵢ is finite then also the subgroup generated by these values, i.e. w(N₁,…,Nᵣ), is finite.