The Lower Central Series of Right Lower-Triangular Nielsen Automorphism Groups

Abstract

Let $F_n$ be a free group of rank $n\ge 3$, freely generated by $x_1,\dots ,x_n$. For $1\le j<i\le n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_i{x}_j$ and $d_{i,j}(x_k)=x_k$ for $k\ne i$, and let $D_n=⟨d_{i,j}\mid 1\le j<i\le n⟩$. We determine the lower central series of $D_n$. We first prove that, for $1\le r<i\le n$, $d_{i,r}\in {\gamma }_{i-r}(D_n)\setminus {\gamma }_{i-r+1}(D_n)$. For each $i=2,\dots ,n$, this calculation leads to a filtration $\{W_{i,c}{\}}_{c\ge 1}$ of $U_i=⟨d_{i,1},\dots ,d_{i,i-1}⟩$. For every $c\ge 1$, we obtain an explicit iterated semidirect-product decomposition of ${\gamma }_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap {\gamma }_c(D_n)=W_{i,c}$ for $i=2,\dots ,n$. The construction also gives an explicit basis for each quotient ${\gamma }_c(D_n)/{\gamma }_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.