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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

5.2 The connection with

5.2 The connection with E( − ,n)Before dealing with the general situation we do the cases d ≤ 2 first:For d = 1 we clearly have S n (1) ∼ = S n for all n ≥ 2.Proposition 5.1 For n ≥ 2 there is an isomorphismE(¡ S n (2) ∼ = , n)s 1 ↦→ (1 2)s 2 ↦→ (1, −1, 0, ..., 0) · (1 2)compatible with the respective split surjections onto S n .PROOF. We consider the following elements of S n (2)a := (2 n)(s 2 (1 2))(2 n) −1 · (1 n)τ k := (k k + 1) k = 1, ..., n − 1.The affine reflection group Ãn−1 has the following presentation, c.f. Section 5.6W (Ãn−1) := 〈α, τ k | τ 2 k , (τ k τ k+1 ) 2 , (τ k τ j ) 2 for |k − j| ≥ 2,α 2 , (ατ 1 ) 3 , (ατ n−1 ) 3 , (ατ k ) 2 for k ≠ 1, n − 1〉.We define a map ˜ϕ : W (Ãn−1) → S n (2) by sending α to a and τ k to τ k for allk. The relations inside W (Ãn−1) also hold true for the corresponding elementsin the image i.e. ˜ϕ extends to a homomorphism. In a similar fashion we definea homomorphism in the opposite direction being the inverse of ˜ϕ. Hence ˜ϕ is anisomorphism.Finally, we identify W (Ãn−1) E(¡ with , n) using the description given inCorollary 3.6 or Example 5.26.□Remark 5.2 There is a general “Coxeter flavour“ in connection with E(−, n).We refer to Section 5.6 for some examples and details.We let F d−1 be the free group of rank d − 1 freely generated by elementsf 2 , ..., f d . We denote by θ the action of S n on F d−1 n given by permuting thefactors. We recall that we constructed E(−, n) using such a θ in Section 3.1.We want to define a mapφ : S n (d) → F n d−1 ⋊ θ S ns 1 ↦→ (1 2)s a ↦→ (f a , f −1 a , 1, ..., 1) · (1 2) ∀a = 2, ..., dϕ(σ) ↦→ σ ∀σ ∈ S nwhere ϕ is the splitting that comes together with S n (d). Since we have fixed thesplitting ϕ of ψ we consider S n as a subgroup of S n (d) and do not mention ϕany further. The content of the following theorem is that this map φ is not only ahomomorphism but also injective with image E(F d−1 , n):44

Theorem 5.3 For n ≥ 5 there exists an isomorphismφ : S n (d) ∼ = E(Fd−1 , n) ≤ F n d−1 ⋊ θ S ns 1 ↦→ (1 2)s a ↦→ (f a , f −1 a , 1, ..., 1) (1 2) ∀a ≥ 2compatible with the respective split surjections onto S n .PROOF. First we have to check that φ extends to a homomorphism. For thiswe only have to check that all relations of S n (d) hold inside the image. Thesecalculations are straight forward and are done in Lemma 5.5.Also, we see from Lemma 5.5 that the image of φ is precisely E(F d−1 , n).Fo a = 2, ..., d and i, j = 1, ..., n we define:ij −1f a := (1, ..., 1, f}{{} a , 1, ..., 1, f a , 1, ..., 1) ∈ F}{{}n d .i.th position j.th positionThese elements generate K(F d−1 , n) as can be seen from applying Lemma 3.1using transpositions as generating set for S n .We want to define a homomorphism from K(F d−1 , n) to S n (d) by sendingˆφ : K(F d−1 , n) → S n (d)iif a ↦→ 1ijf a ↦→ (1 i)(2 j) · (s a (1 2)) · (2 j) −1 (1 j) −1 i ≠ jFrom Proposition 5.6 we know all the relations that hold between the f a ij insideK(F d−1 , n). The relations (∗2) and (∗3) hold true in S n (d) by the relations comingfrom cuspidal transpositions. The relations (∗4) hold true because of the relationscoming from nodal transpositions. We leave the details to the reader.By definition φ is the identity when restricted to S n . To show that ˆφ extendsto a homomorphism from K(F d−1 , n) ⋊ θ S n to S n (d) we only have to show thatˆφ is S n -equivariant with respect to the S n -action given by conjugation in bothgroups. We leave it to the reader to show that for σ ∈ S nσ · f aij · σ −1= f aσ −1 (i) σ −1 (j)σ · (1 i)(2 j) · (s a (1 2)) · (2 j) −1 (1 j) −1 · σ −1= (1 σ −1 (i))(2 σ −1 (j)) · (s a (1 2)) · (2 σ −1 j) −1 (1 σ −1 (j)) −1holds true proving S n -equivariance.Hence, there is a homomorphism from E(F d−1 , n) to S n (d) prolonging ˆφ andcompatible with the split surjections onto S n . Since φ is surjective and ˆφ◦φ(s a ) =s a for all a it follows that φ is an isomorphism.□45

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