Views
3 years ago

# 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

The Path Ahead: Fundamentals of Generative ... - Right 4 Your Type
Project Management Fundamentals
Generic process improvement approach applied to IT projects - PMI ...
Project Insight - Power Generation Construction Projects in Asia-Pacific
Groups of integral representation type - MSP
Carbon reductions generate positive ROI - Carbon Disclosure Project
Renewable Projects Portal for CLP Group - OSIsoft
Building Next Generation Design Support Programmes - See Project
Projected Costs of Generating Electricity - OECD Nuclear Energy ...
GALOIS THEORY - Tata Institute of Fundamental Research
On The Structure of Certain Galois Cohomology Groups
The Fundamental Theorem of Galois Theory and Normal Subgroups
Generic Galois groups for q-difference equations
fields and galois theory - Neil Strickland - University of Sheffield
periods of eisenstein series: the galois case - Project Euclid
Introduction to Differential Galois Theory
Galois module structure of local unit groups
Supersingular Galois representations and a generalization of a ...
Galois groups with restricted ramification Romyar Sharifi Harvard ...
On Galois Algebras with a Unique Galois Group - Bradley Bradley
Chapter 4 COVERING PROJECTIONS AND FUNDAMENTAL GROUP
On Galois Algebras with an Inner Galois Group - Bradley Bradley
QUANTUM GALOIS THEORY FOR FINITE GROUPS
Automorphy lifting for residually reducible l-adic Galois ...
ON GALOIS EXTENSIONS WITH AN INNER GALOIS GROUP ... - FUJI
Wildly ramified Galois representations and a generalization of a ...
Model Theory of Absolute Galois Groups - FreiDok - Albert-Ludwigs ...
Connected Linear Groups as Differential Galois Groups (with C ...
1 The Galois Group of a Quadratic 2 The Galois Group of a Cubic
A family of polynomials with Galois group \$ PSL_5 (2) \$ over ...