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.
Funding Statement
The authors are grateful to the anonymous referee for useful comments, which improve the present paper. The second author is supported by JSPS KAKENHI Grant Number JP20H00114 and JST-Mirai Program Grant Number JPMJMI22G1. The third author is supported by JSPS KAKENHI Grant Number JP21J11199. The first author, the fourth author and the fifth author are partially supported by JSPS KAKENHI Grant Number JP21K13790, JP19K14536 and JP21K03241, respectively.
Citation
Morimichi Kawasaki. Mitsuaki Kimura. Shuhei Maruyama. Takahiro Matsushita. Masato Mimura. "Mixed commutator lengths, wreath products and general ranks." Kodai Math. J. 46 (2) 145 - 183, June 2023. https://doi.org/10.2996/kmj46202
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