Mixed commutator lengths, wreath products and general ranks

TL;DR

This paper investigates mixed commutator lengths in wreath products, linking them to the general rank, and explores conditions under which these lengths coincide or differ, especially focusing on properties of the group 3.

Contribution

It determines mixed commutator lengths in wreath products via the general rank and characterizes when these lengths match or differ based on group properties.

Findings

Mixed commutator lengths are expressed in terms of the general rank for wreath products.

When 3 is not locally cyclic, cl_G and cl_{G,N} differ on [G,N].

If 3 is locally cyclic, cl_G and cl_{G,N} coincide on [G,N] for certain extensions.

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.