论文标题
混合换向器长度，花圈产品和一般排名
Mixed commutator lengths, wreath products and general ranks
论文作者
论文摘要
在本文中，对于一对$ g $的一对$（g，n）$及其普通子组$ n $，我们认为混合换向器子组$ [g，n] $上的混合换向器长度$ \ mathrm {cl} _ {g，n} $。我们专注于花圈产品的设置：$（g，n）=（\ mathbb {z} \wrγ，\bigoplus_γ\ mathbb {z}）$。然后，我们根据MALCEV的意义确定一般等级的混合换向器长度。作为副产品，当Abelian Group $γ$不是本地循环时，普通的换向器长度$ \ Mathrm {Cl} _g $与上述对的$ [G，N]上的$ [G，N} $不一致与$ \ Mathrm {Cl} _ {g，n} $。另一方面，我们证明，如果$γ$是本地循环的，那么对于每对$（g，n）$，因此$ 1 \ to n \ to n \ to g \ to g \至γ\至1 $是精确的，$ \ mathrm {cl} _ {g} _ {g} $ and $ \ \ \ \ \ \ mathrm {cl} $ n} $ co，当组$γ$属于与表面基团相关的某些类别时，我们还研究了排列花圈产品的情况。
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 Γ, \bigoplus_Γ\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 $Γ$ 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 $Γ$ is locally cyclic, then for every pair $(G,N)$ such that $1\to N\to G\to Γ\to 1$ 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 $Γ$ belongs to a certain class related to surface groups.
