Artin automorphisms of the free group

Let $F_n=⟨x_1,\dots ,x_n⟩$ be the free group of Free group on a set of generators, identified with ${\pi }_1(D^2\setminus Q_n,d)$ through the standard meridians by The punctured-disk fundamental group is free on the standard meridians. For $1\le i\le n-1$, $\rho ({\sigma }_i)\in \mathrm{Aut}(F_n)$ is the automorphism given on the basis by $\rho ({\sigma }_i)(x_i)=x_i{x}_{i+1}{x}_i^{-1},\rho ({\sigma }_i)(x_{i+1})=x_i,\rho ({\sigma }_i)(x_j)=x_j(j\mathrm{\notin }\{i,i+1\})\mathrm{.}$ Its inverse is $\rho ({\sigma }_i{)}^{-1}(x_i)=x_{i+1},\rho ({\sigma }_i{)}^{-1}(x_{i+1})=x_{i+1}^{-1}{x}_i{x}_{i+1},\rho ({\sigma }_i{)}^{-1}(x_j)=x_j(j\mathrm{\notin }\{i,i+1\})\mathrm{.}$ Products use ordinary function composition: the leftmost factor is the outermost map and the rightmost factor acts first. Also $\rho ({\sigma }_i^{-1}):=\rho ({\sigma }_i{)}^{-1}$.

The assignments are automorphisms, and the displayed formulas are inverse. By the universal property of the free group (Free group on a set of generators, Reduced words form the free group on an alphabet) the displayed values on the free basis extend to a unique endomorphism $\rho ({\sigma }_i):F_n\to F_n$, and likewise the displayed inverse formulas extend to an endomorphism $\theta :F_n\to F_n$. Substituting, $\theta (\rho ({\sigma }_i)(x_i))=\theta (x_i{x}_{i+1}{x}_i^{-1})=\theta (x_i)\theta (x_{i+1})\theta (x_i{)}^{-1}=x_{i+1}(x_{i+1}^{-1}{x}_i{x}_{i+1})x_{i+1}^{-1}=x_i,$ $\theta (\rho ({\sigma }_i)(x_{i+1}))=\theta (x_i)=x_{i+1},$ and both composites fix every other basis element; so $\theta \circ \rho ({\sigma }_i)={\mathrm{id}}_{F_n}$ on a basis, hence as endomorphisms. Symmetrically $\rho ({\sigma }_i)\circ \theta ={\mathrm{id}}_{F_n}$. Therefore $\rho ({\sigma }_i)$ is a bijection with the displayed inverse, i.e. an automorphism (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).

Convention note. The displayed formulas are Artin's equations (14) and (15) with the letter ${\sigma }_i$ and its inverse interchanged; under the frozen geometric conventions of this library the positive half twist is the anticlockwise supported half rotation (see def-elementary-geometric-half-twist and thm-braid-group-is-the-boundary-fixed-mapping-class-group-of-the-punctured-disk), and prop-the-geometric-action-on-meridians-is-the-artin-representation proves that its action on the standard meridians is exactly the substitution frozen above. The interchange is a convention, not a change of mathematical content: $\rho$ and Artin's substitution generate the same subgroup of $\mathrm{Aut}(F_n)$ and satisfy the same braid relations, and the companion page shows that the mirror (clockwise) half rotation realizes Artin's displayed formulas verbatim.

• For $n=1$ there is no index $i$ with $1\le i\le n-1$, so there is no Artin automorphism and the statements making use of them are vacuous.
• $\rho ({\sigma }_i)$ fixes $x_j$ for all $j\mathrm{\notin }\{i,i+1\}$: its support is the pair of adjacent letters. This is the algebraic shadow of the fact that the half twist is supported in the disc $U_i$ around the adjacent punctures.