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Author: Pierre Guillot
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Description: Sage script by P. Guillot

Computation of GT1(G)\mathcal{GT}_1(G)

Pierre Guillot

This computes the group GT1(G)\mathcal{GT}_1(G), described in my paper The Grothendieck-Teichmüller group of a finite group and GG-dessins d'enfants.

In order to use and modify the script, you first need to:

  • create an accound on SageMathCloud, if you haven't got one already,
  • create at least one "project" on SageMathCloud.

Then you can click on "Copy Jupyter Notebook to your project...", in the top left hand corner.

Alternatively, if you have a recent version of Sage installed, with the Jupyter Notebook working, you can click on "Download".

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IsContained= libgap.function_factory("\in")
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class GT1: def __init__(self, G, verbose= False, very_verbose= False): self.G= G self.verbose= verbose or very_verbose self.very_verbose= very_verbose def compute_cover_group(self): if self.verbose: print "*** computing cover" ##product group GG= self.G.DirectProduct(self.G) i1= GG.Embedding(1) i2= GG.Embedding(2) p1= GG.Projection(1) p2= GG.Projection(2) ##automorphism group of G autG= self.G.AutomorphismGroup() #convert the elements of autG into automorphisms of GG gensautG= autG.GeneratorsOfGroup() gensGG= GG.GeneratorsOfGroup() imgs= [] for phi in gensautG: phi_on_gens= [ (p1*phi*i1).Image(gh) * (p2*phi*i2).Image(gh) for gh in gensGG ] phiGG= GG.GroupHomomorphismByImages(GG, gensGG, phi_on_gens) imgs.append(phiGG) #using this, embed autG in autGG... diagautG= libgap.Group(imgs) ### classes of pairs in G classes= diagautG.OrbitsDomain(GG) if self.very_verbose: print len(classes), "classes of pairs" def is_generating_pair(C): pair= C[0] x= p1.Image(pair) y= p2.Image(pair) return self.G.Subgroup([x, y]) == self.G gens_classes= [ C for C in classes if is_generating_pair(C) ] r= len(gens_classes) if self.very_verbose: print r,"of them generate G" ###build G^r and then GB Gr= libgap.DirectProduct([ self.G for i in range(r) ]) inj= [ Gr.Embedding(i) for i in [1..r] ] self.x= prod([ (p1*inj[i]).Image(gens_classes[i][0]) for i in range(r) ]) self.y= prod([ (p2*inj[i]).Image(gens_classes[i][0]) for i in range(r) ]) self.GB= Gr.Subgroup([self.x, self.y]) def compute_automorphisms(self): if self.verbose: print "*** computing representatives for GT1" # for convenience GB= self.GB x= self.x y= self.y one= GB.Identity() ### the centralizers Cx= GB.Centralizer(x) Cy= GB.Centralizer(y) ### find the double cosets doubles= GB.DoubleCosets(Cx, Cy) if self.very_verbose: print "found", len(doubles), "double cosets" wait= len(doubles) ### prepare theta and delta theta= GB.GroupHomomorphismByImages(GB, [x, y], [y, x]) delta= GB.GroupHomomorphismByImages(GB, [x, y], [y^(-1)*x^(-1), y]) ### do all the tests ! self.auts= [] for i, C in enumerate(doubles): if self.very_verbose: print "coset", i+1, "out of", wait #take a representative c= C.Representative() xp= x^c #first test: <x^c, y> is all of GB GBp= GB.Subgroup([xp, y]) if GBp != GB: continue # try next C #second test: commutation with theta ct= c^theta if not IsContained(one, Cx.DoubleCoset(c*ct, Cy^ct)): continue #third test: automorphism is actually well-defined phi= GB.GroupHomomorphismByImages(GB, [x, y], [xp, y]) if phi == libgap.eval("fail"): continue #fourth test: commutation with delta if not libgap.IsInnerAutomorphism( phi*delta*phi^(-1)*delta^(-1)): continue #if we're here, all the tests have been passed! if self.very_verbose: print "found an element in GT1(G) !" self.auts.append(phi)
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#G= libgap.SmallGroup(64,34) G= libgap.PSL(2,2)
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gt= GT1(G, very_verbose= True)
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*** computing cover 11 classes of pairs 3 of them generate G
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*** computing representatives for GT1 found 3 double cosets coset 1 out of 3 found an element in GT1(G) ! coset 2 out of 3 found an element in GT1(G) ! coset 3 out of 3
In [9]:
[[ (2,3)(5,6)(7,8,9), (1,2)(4,5,6)(8,9) ] -> [ (2,3)(5,6)(7,8,9), (1,2)(4,5,6)(8,9) ], [ (2,3)(5,6)(7,8,9), (1,2)(4,5,6)(8,9) ] -> [ (2,3)(5,6)(7,9,8), (1,2)(4,5,6)(8,9) ]]
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