Peripheral-boundary-preserving automorphisms of F_n

Let $n\in \mathbb{N}$ and let $F_n=⟨x_1,\dots ,x_n⟩$ be the free group of Free group on a set of generators on the $n$ letters $x_1,\dots ,x_n$, with reduced words and free reduction as in Reduced words form the free group on an alphabet. The ordered boundary product is the element $\delta :=x_1{x}_2\cdots x_n\in F_n,$ the word $x_1\cdots x_n$; under the identification of $x_i$ with the class of the standard meridian of Standard meridians of a punctured disk it is the element represented by the positively oriented boundary loop (proved in lem-the-oriented-boundary-loop-represents-the-ordered-product-of-the-standard-meridians).

An automorphism $A\in \mathrm{Aut}(F_n)$ (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$) is peripheral-boundary-preserving if

1. (peripheral) for every $i$ the element $A(x_i)$ is conjugate in $F_n$ to one of the generators $x_1,\dots ,x_n$; equivalently, after rewriting $A(x_i)$ in the reduced normal form of Reduced words form the free group on an alphabet, there are a permutation $\pi$ of $\{1,\dots ,n\}$ and a reduced word $Q_i$ with $A(x_i)=Q_i^{-1}{x}_{\pi (i)}{Q}_i;\text{and}$
2. (boundary-preserving) $A$ fixes the ordered boundary product, $A(x_1{x}_2\cdots x_n)=x_1{x}_2\cdots x_n\mathrm{.}$

These are exactly the two hypotheses of Artin's characterization of the braid subgroup of $\mathrm{Aut}(F_n)$.

The two formulations of condition 1 agree. If $A(x_i)=Q_i^{-1}{x}_{\pi (i)}{Q}_i$, its class in the abelianisation $F_n^{\mathrm{a}\mathrm{b}}\cong {\mathbb{Z}}^n$ is the class of $x_{\pi (i)}$; conversely, an automorphism induces an automorphism of ${\mathbb{Z}}^n$, so if every $A(x_i)$ is conjugate to a generator, the assignment $e_i\mapsto e_{\pi (i)}$ is an invertible self-map of the basis and $\pi$ is a permutation. The element $Q_i$ is not unique, but it is unique up to left-multiplication by powers of the middle generator: if $Q^{-1}{x}_{j}Q=R^{-1}{x}_{j}R$ then $R{Q}^{-1}$ commutes with $x_j$, hence lies in the centraliser $⟨x_j⟩$, so $R=x_j^{m}Q$ for some integer $m$; thus the invariant content of condition 1 is "$A(x_i)$ is conjugate to a generator", and the displayed form is a normalised way of writing that conjugacy. Condition 2 fixes the ordered product itself, not merely its conjugacy class or its image in the abelianisation. No choice principle is used in this definition.

• For $n\le 1$ the group $F_n$ is trivial or infinite cyclic and the conditions are checked directly; the ordered product is $x_1$ for $n=1$.
• For $n\ge 2$ the conditions are independent. The basis transposition $x_1\leftrightarrow x_2$ preserves peripheral conjugacy classes and changes $\delta$. Conversely, the substitution $x_1\mapsto x_1^{-1}$, $x_2\mapsto x_1^2{x}_2$, fixing the other generators, fixes $\delta$ and is an involution, hence an automorphism. Its image of $x_1$ has abelianised class $-e_1$, so it is not conjugate to any positive basis generator.