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Artin automorphisms permute meridian conjugacy classes and fix the boundary word

Statement

For every braid word $β$ in the generators $σ_{1}^{\pm 1}, \dots, σ_{n - 1}^{\pm 1}$ and every $i$ , the element $ρ (β) (x_{i})$ is conjugate in $F_{n}$ to one of the generators $x_{1}, \dots, x_{n}$ , and $ρ (β) (x_{1} x_{2} \dots x_{n}) = x_{1} x_{2} \dots x_{n} .$ Here $x_{1} \dots x_{n}$ is the boundary word, i.e. the element represented by the positively oriented boundary loop $\partial$ by The oriented boundary loop represents the ordered product of the standard meridians. No choice principle is used.

Facts & Assumptions

Given: the free group $F_{n} = ⟨ x_{1}, \dots, x_{n} ⟩$ with its reduced words, the generators $ρ (σ_{i})^{\pm 1}$ of Artin automorphisms of the free group, the homomorphism $ρ : B_{n} \to Aut (F_{n})$ of The Artin representation on a free group, an arbitrary braid word $β$ , and the boundary loop $\partial$ with its class of The oriented boundary loop represents the ordered product of the standard meridians.

[F1]

The generator substitutions. For $1 \leq i \leq n - 1$ , $ρ (σ_{i}) (x_{i}) = x_{i} x_{i + 1} x_{i}^{- 1}, ρ (σ_{i}) (x_{i + 1}) = x_{i}, ρ (σ_{i}) (x_{j}) = x_{j} (j \notin {i, i + 1}),$ so $ρ (σ_{i})$ sends $x_{i}$ to the conjugate $x_{i} x_{i + 1} x_{i}^{- 1}$ of $x_{i + 1}$ , sends $x_{i + 1}$ to the generator $x_{i}$ , and fixes every other generator; and $ρ (σ_{i}) (x_{i} x_{i + 1}) = (x_{i} x_{i + 1} x_{i}^{- 1}) x_{i} = x_{i} x_{i + 1},$ so $ρ (σ_{i})$ fixes the ordered product. The inverse $ρ (σ_{i})^{- 1}$ has image formulas $ρ (σ_{i})^{- 1} (x_{i}) = x_{i + 1}$ , $ρ (σ_{i})^{- 1} (x_{i + 1}) = x_{i + 1}^{- 1} x_{i} x_{i + 1}$ and fixes all other generators, so it too carries every basis letter to a conjugate of a generator and fixes the ordered product. (Artin automorphisms of the free group.)

[F2]

The representation. $ρ$ is a group homomorphism, so $ρ (β)$ is the composite of the automorphisms attached to the letters of $β$ , with the leftmost letter the outermost map (the rightmost map is evaluated first), and $ρ$ of the empty word is the identity. Two endomorphisms of $F_{n}$ agree as soon as they agree on the free basis $x_{1}, \dots, x_{n}$ , and equality of elements is decided by reduced words (The Artin representation on a free group, Free group on a set of generators, Reduced words form the free group on an alphabet).

[F3]

The boundary word. The class of the loop $x_{1} \dots x_{n}$ is $[x_{1}] \dots [x_{n}]$ , and under the identification of $π_{1} (D^{2} ∖ Q_{n}, d)$ with $F_{n}$ by the standard meridians it corresponds to the positively oriented boundary loop $\partial$ (The oriented boundary loop represents the ordered product of the standard meridians, Standard meridians of a punctured disk).

Proof

Proof technique: direct, by generators and preservation under composition and inversion.

1.1F1

The generator substitutions have the two properties. For each $i$ and each sign, $ρ (σ_{i})^{\pm 1}$ carries every basis letter $x_{j}$ to a conjugate of a generator: by [F1] the values are unchanged generators, $x_{i} x_{i + 1} x_{i}^{- 1}$ , or $x_{i + 1}^{- 1} x_{i} x_{i + 1}$ , all of which are conjugates of generators (a generator is conjugate to itself via the empty word). Moreover $ρ (σ_{i})^{\pm 1}$ fixes the ordered product $δ = x_{1} \dots x_{n}$ : for $ρ (σ_{i})$ this is the last display of [F1], and for the inverse it follows by applying $ρ (σ_{i})^{- 1}$ to the equality $ρ (σ_{i}) (δ) = δ$ and using $ρ (σ_{i})^{- 1} ρ (σ_{i}) = id$ .

1.2F1algebra

The two properties are preserved by composition and inversion. Let $A, B \in Aut (F_{n})$ satisfy: $A (x_{j})$ and $B (x_{j})$ are conjugate to generators for every $j$ , and $A (δ) = B (δ) = δ$ . For the composite $(A \circ B)$ , write $B (x_{j}) = Q^{- 1} x_{k} Q$ with $Q \in F_{n}$ ; then $(A \circ B) (x_{j}) = A (Q)^{- 1} A (x_{k}) A (Q),$ a conjugate of $A (x_{k})$ , which is a conjugate of a generator; and $(A \circ B) (δ) = A (B (δ)) = A (δ) = δ$ . For the inverse, abelianisation sends each basis vector $e_{k}$ to some $e_{π (k)}$ ; since the induced map is invertible, $π$ is a permutation. Thus for every $j$ there is a $k$ with $A (x_{k}) = Q^{- 1} x_{j} Q$ ; applying $A^{- 1}$ and rearranging gives $A^{- 1} (x_{j}) = A^{- 1} (Q) x_{k} A^{- 1} (Q)^{- 1},$ a conjugate of a generator, and $A^{- 1} (δ) = δ$ because $A (δ) = δ$ .

2.1F2step 1.1step 1.2

Induction on the letters of the word. Let $β$ be a braid word $β_{1} β_{2} \dots β_{m}$ with letters $β_{k} \in {σ_{1}^{\pm 1}, \dots, σ_{n - 1}^{\pm 1}}$ . If $m = 0$ , then $β$ is the empty word and $ρ (β) = id$ , for which $ρ (β) (x_{j}) = x_{j}$ is a conjugate of a generator and $ρ (β) (δ) = δ$ . If $m \geq 1$ , write $β = β_{1} β^{'}$ ; by [F2] $ρ (β) = ρ (β_{1}) \circ ρ (β^{'})$ , where $ρ (β_{1})^{\pm 1}$ is one of the automorphisms of step 1.1 and, by induction on $m$ , $ρ (β^{'})$ carries every basis letter to a conjugate of a generator and fixes $δ$ . Step 1.2 applied to $A = ρ (β_{1})$ and $B = ρ (β^{'})$ then gives both properties for $ρ (β)$ .

3.1F3step 2.1∎

Conclusion. Steps 1.1, 1.2 and 2.1 show that every braid word $β$ induces an automorphism carrying each $x_{i}$ to a conjugate of a generator and fixing $δ = x_{1} \dots x_{n}$ ; by [F3] this element is the one represented by the boundary loop $\partial$ , which proves the statement. Every verification above was a finite computation with the displayed substitutions, so no choice principle is used.

Remarks

The two properties are exactly the necessary conditions of Artin's characterization of the braid subgroup of $Aut (F_{n})$ : see thm-artins-characterization-of-the-braid-subgroup-of-aut-f-n.
Only the direction from the word to the automorphism is asserted here; the converse, that every automorphism with the two properties comes from a braid word, is thm-every-peripheral-boundary-preserving-free-group-automorphism-is-an-artin-automorphism.

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Sources

Juan Gonzalez-Meneses, Basic results on braid groups, section 1.6, printed pp. 8-10 (each rho_beta(x_j) is a conjugate of a generator and rho_beta(x_1...x_n) = x_1...x_n) (standard reference, not scraped)
Emil Artin, Theory of Braids, Annals of Mathematics 48 (1947), pp. 101-126, equation (13) and Theorem 15, printed pp. 112-113 (standard reference, not scraped)