Mixed commutator lengths, wreath products and general ranks

Abstract: In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $\mathrm{cl}{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(\mathbb{Z}\wr \Gamma, \bigoplus{\Gamma}\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $\Gamma$ is not locally cyclic, the ordinary commutator length $\mathrm{cl}G$ does not coincide with $\mathrm{cl}{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if $\Gamma$ is locally cyclic, then for every pair $(G,N)$ such that $1\to N\to G\to \Gamma \to 1$ is exact, $\mathrm{cl}{G}$ and $\mathrm{cl}{G,N}$ coincide on $[G,N]$. We also study the case of permutational wreath products when the group $\Gamma$ belongs to a certain class related to surface groups.