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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Remark 5.4 The really

Remark 5.4 The really hard part of this proof is Proposition 5.6. It says thatthe relations of K(F d , n) are only some “obvious“ ones, i.e. a certain set ofcommutator relations.The author’s original proof used a Reidemeister-Schreier rewriting process toobtain a presentation of the subgroup E(F d−1 , n) of F d−1 n ⋊ θ S n . However, sincethe subgroup has infinite index in the ambient group he obtained an infinite set ofrelations. The computations were a ten page flow of quite messy calculations.Meanwhile, [RTV] appeared and the author decided to copy their proof.Lemma 5.5 Let G be an arbitrary group and ⃗g i , i = 1, 2 two elements of K(G, n).We defines i := ⃗g i (1 2)⃗g i −1 , i = 1, 2Then the following relations hold inside E(G, n)2s i i = 1, 2(s i · τ) 2 if τ and (1 2) are nodal transpositions(s i · τ) 3 if τ and (1 2) are cuspidal transpositions(σs i σ −1 · s j ) 2 if σ(1 2)σ −1 and (1 2) are nodal transpositions(σs i σ −1 · s j ) 3 if σ(1 2)σ −1 and (1 2) are cuspidal transpositions.If n ≥ 3 and if the elements g 1 , ..., g s generate G then E(G, n) is generated by[(g i , 1, ..., 1), (1 2)] and an arbitrary generating set of S n .PROOF. The first relation is straight forward from Lemma 3.1. Furthermore itallows us to view the remaining relations as commutator relations or triple commutatorrelations, respectively.We do the computations inside G n ⋊ S n as usual. We set τ = (3 4) and⃗g = (g 1 , g 2 , ..., g n ) ∈ G n , and check that ⃗g(1 2)⃗g −1 and τ commute:((⃗g(1 2)⃗g −1 ) · τ) 2= [⃗g(1 2)⃗g −1 , τ]= ⃗g(1 2)⃗g −1 · τ ( (g 1 g −12 , g 2 g −11 , 1, ..., 1) −1 (1 2) −1) τ −1= ⃗g(1 2)⃗g −1 · τ ( (g 1 g −12 , g 2g −11 , 1, ..., 1)−1) τ −1 (1 2) −1= ⃗g(1 2)⃗g −1 · (g 1 g −12 , g 2 g −11 , 1, ..., 1) −1 (1 2) −1= ⃗g(1 2)⃗g −1 · (⃗g(1 2)⃗g −1 ) −1= 1We leave the remaining relations to the reader.We have already seen in Lemma 3.1 that E(G, n) is generated by S n and allelements of the form (g, g −1 , 1, ..., 1). Let g 1 , ..., g s be a generating set for G. Wedefine ⃗g i := (g i , 1, ..., 1) and compute for n ≥ 3[⃗g i , (1 3)] · [⃗g j , (1 2)] · [⃗g i , (1 3)] = (g i g j , (g i g j ) −1 , 1, ..., 1)So we get all elements (g, g −1 , 1, ...., 1) from the set [⃗g i , (1 2)] and S n .46□

Proposition 5.6 (Rowen, Teicher, Vishne) We let F d be the free group of rank dand assume that it is freely generated by elements f 1 , ..., f d . We set:ij −1nf a := (1, ..., 1, f }{{} a , 1, ..., 1, f a , 1, ..., 1) ∈ F}{{}di.th position j.th positionIf n ≥ 2 then K(F d , n) is generated by f a ij with a = 1, ..., d and i, j = 1, ..., n.And if n ≥ 5 then all relations inside K(F d , n) follow from the following relations:iif a = 1 (∗1)ij jkf a · f a =ikf a (∗2)ik ij ikf[ a · f a = f a (∗3)fa , f ] kl b = 1 if i, j, k, l are all different. (∗4)In other words we have a finite presentation of K(F d , n) for n ≥ 5.PROOF. The proof is taken from [RTV, Theorem 5.7]. However, we adapted thenotations to our situation.First of all, the f ij a ’s generate K(F d , n). This follows from Lemma 3.1 appliedto the generating set f i of F d and taking as generating set for S n the set of alltranspositions.We leave it to the reader to show that the relations given in the statement ofProposition 5.6 hold true in F n d and hence in K(F d , n).We define K d,n to be the group generated by elements f ij a with a = 1, ..., dand i, j = 1, ..., n subject to the relations given by Proposition 5.6. We haveshown above that there is a surjective homomorphism from K d,n onto K(F d , n).Next, we define Kd,n ∗ to be the group generated by elementsf aijand t a with a = 1, .., d, i, j = 1, ..., nsubject to the relations of K d,n and the relations[t a , f ij b ] = [ f nk a , f ] ij b k ≠ i, j (†1)[t a , t b ] = [f ni a , f nj b ] i ≠ j and i, j ≠ n (†2)Then we define the following mapµ : K ∗ d,n → F dnt a ↦→ f anf aij↦→ (f a j ) −1 f aiwhere f a i denotes the element (1, ..., 1, f a , 1, ..., 1) of F d n having its non-trivialentry in the i.th position. By Lemma 5.7 this map µ defines an isomorphism ofgroups.47

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