下三角Nielsen自同构群的下中心列
The Lower Central Series of Right Lower-Triangular Nielsen Automorphism Groups
AI总结：
本文确定秩$n\geq3$自由群的下三角Nielsen自同构群$D_n$的下中心列，推导其滤链、半直积分解及商群基，证明$D_n$是Magnus群。
AI中文摘要：
设$F_n$是秩为$n\geq3$的自由群，由$x_1,\ldots,x_n$自由生成。对$1\leq j<i\leq n$，记$d_{i,j}$为$F_n$的Nielsen自同构，定义为$d_{i,j}(x_i)=x_ix_j$，当$k\neq i$时$d_{i,j}(x_k)=x_k$；令$D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$。本文确定$D_n$的下中心列：首先证明对$1\leq r<i\leq n$，$d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$；对每个$i=2,\ldots,n$，该计算导出$U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$的滤链$\{W_{i,c}\}_{c\geq1}$；对每个$c\geq1$，得到$\gamma_c(D_n)$关于子群$W_{i,c}$的显式迭代半直积分解，且证明$U_i\cap\gamma_c(D_n)=W_{i,c}$（$i=2,\ldots,n$）；该构造还给出每个商群$\gamma_c(D_n)/\gamma_{c+1}(D_n)$关于基本换位子的显式基，并确定每个此类换位子的精确下中心深度，进而证明$D_n$是Magnus群。
英文摘要：
Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in γ_{i-r}(D_n)\setminusγ_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $γ_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\capγ_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $γ_c(D_n)/γ_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.
