Presentations of Lower-Triangular Subgroups of $\mathrm{Aut}(F_n)$

Abstract

Let $F_n$ be the free group of rank $n\ge 2$ with basis $x_1,\dots ,x_n$. For $1\le j<i\le n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_i{x}_j$ and $e_{i,j}(x_i)=x_j{x}_i$, respectively, and fixing the remaining free generators. Write $D_n=⟨d_{i,j}\mid 1\le j<i\le n⟩$ and $A_n^{+}=⟨d_{i,j},e_{i,j}\mid 1\le j<i\le n⟩$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\le r\le n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^{+}$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\le r\le n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^{+}$ may be chosen so that every defining relator is a single commutator.