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: $(G,N)=(\mathbb{Z}\wr \hat{I}“,{⨁}_{\hat{I}}“\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $\hat{I}“$ 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 $\hat{I}“$ is locally cyclic, then for every pair $(G,N)$ such that $1\to N\to G\to \hat{I}“\to 1$ 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 $\hat{I}“$ belongs to a certain class related to surface groups.

33 pages, no figure, Proposition 4.4 in the previous version was false, and we replaced it with a weaker version. To appear in Kodai Mathematical Journal