chap4.txt

  
  4 Farey symbols for congruence subgroups
  
  The  package  Congruence  provides  functions to construct Farey symbols for
  finite  index  subgroups.  The  algorithm  used  in  the  package  allows to
  construct  a Farey symbol for any finite index subgroup of SL_2(ℤ) for which
  it  is  possible to check whether a given matrix belongs to this subgroup or
  not.
  
  The development of an algorithm to determine the Farey symbol for a subgroup
  G  of  a finite index in SL_2(ℤ) was started by Ravi Kulkarni in [Kul91] and
  later  it  was improved by Mong-Lung Lang, Chong-Hai Lim and Ser-Peow Tan in
  [LLT95b], [LLT95a].
  
  
  4.1 Computation of the Farey symbol for a finite index subgroup
  
  4.1-1 FareySymbol
  
  FareySymbol( G )  attribute
  
  For  a  subgroup of a finite index G, this attribute stores one of the Farey
  symbols  corresponding  to  the congruence subgroup G. The algorithm for its
  computation  will  work  for any matrix group for which a membership test is
  available.
  
    Example  
    
    gap> FareySymbol(PrincipalCongruenceSubgroup(8));
    [ infinity, 0, 1/4, 1/3, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3, 3/4, 1, 5/4, 4/3, 
      11/8, 7/5, 3/2, 8/5, 13/8, 5/3, 7/4, 2, 9/4, 7/3, 19/8, 12/5, 5/2, 13/5, 
      21/8, 8/3, 11/4, 3, 13/4, 10/3, 27/8, 17/5, 7/2, 18/5, 29/8, 11/3, 15/4, 4, 
      17/4, 13/3, 9/2, 14/3, 19/4, 5, 21/4, 16/3, 11/2, 17/3, 23/4, 6, 25/4, 
      19/3, 13/2, 20/3, 27/4, 7, 29/4, 22/3, 15/2, 23/3, 31/4, 8, infinity ]
    [ 1, 17, 10, 26, 32, 18, 19, 27, 30, 5, 2, 2, 13, 28, 26, 20, 21, 29, 27, 7, 
      3, 3, 16, 31, 28, 22, 23, 33, 29, 9, 4, 4, 5, 30, 31, 24, 25, 32, 33, 12, 
      6, 6, 7, 19, 18, 15, 8, 8, 9, 21, 20, 10, 11, 11, 12, 23, 22, 13, 14, 14, 
      15, 25, 24, 16, 17, 1 ]
    gap> FareySymbol(CongruenceSubgroupGamma0(20));
    [ infinity, 0, 1/5, 1/4, 2/7, 3/10, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1, 
      infinity ]
    [ 1, 3, 4, 6, 7, 7, 5, 2, 2, 3, 6, 4, 5, 1 ]  
    
  
  
  
  4.2  Computation  of  generators  of  a finite index subgroup from its Farey
  symbol
  
  If fs is the Farey symbol for a group G with r_1 even labels, r_2 odd labels
  and  r_3  pairs  of  intervals, then G is generated by r_1+r_2+r_3 matrices,
  which  form  a  set  of  independent  generators  for  G. These matrices are
  constructed as follows:
  
  for each even interval [x_i, x_i+1], take the matrix
  
                         A=  [a_{i+1} b_{i+1} + a_i b_i    -a_i^2 - a_{i+1}^2        ]
                             [b_i^2 +b_{i+1}^2             -a_{i+1} b_{i+1} - a_i b_i]
  
  for each odd interval [x_j,x_j+1], take the matrix
  
                          B=  [a_{j+1} b_{j+1} + a_j b_{j+1} + a_j b_j      -a_j^2 - a_j a_{j+1} -a_{j+1}^2]
                              [ b_j^2 + b_j b_{j+1} + b_{j+1}^2  -a_{j+1}   b_{j+1} - a_{j+1} b_j - a_j b_j]
  
  for each pair of free intervals [x_k,x_k+1] and [x_s,x_s+1], take the matrix
  
                          C=  [a_{s+1} b_{k+1} + a_s b_k    -a_s a_k - a_{s+1} a_{k+1}]
                              [b_s b_k- b_{s+1} b_{k+1}c    -a_{k+1} b_{s+1} - a_k b_s]
  
  4.2-1 MatrixByEvenInterval
  
  MatrixByEvenInterval( gfs, i )  function
  
  Returns  the  matrix corresponding to the even interval i in the generalized
  Farey sequence gfs.
  
    Example  
    
    gap> H:=CongruenceSubgroupGamma0(5); 
    <congruence subgroup CongruenceSubgroupGamma_0(5) in SL_2(Z)>
    gap> fs:=FareySymbol(H);
    [ infinity, 0, 1/2, 1, infinity ]
    [ 1, "even", "even", 1 ]
    gap> gfs:=GeneralizedFareySequence(fs);
    [ infinity, 0, 1/2, 1, infinity ]
    gap> MatrixByEvenInterval(gfs,2);      
    [ [ 2, -1 ], [ 5, -2 ] ]
    
  
  
  4.2-2 MatrixByOddInterval
  
  MatrixByOddInterval( gfs, i )  function
  
  Returns  the  matrix  corresponding to the odd interval i in the generalized
  Farey sequence gfs.
  
    Example  
    
    gap> fs_oo:=FareySymbolByData([infinity,0,infinity],["odd","odd"]);;
    gap> gfs_oo:=GeneralizedFareySequence(fs_oo);
    [ infinity, 0, infinity ]
    gap> MatrixByOddInterval(gfs_oo,1);
    [ [ -1, -1 ], [ 1, 0 ] ]
    
  
  
  4.2-3 MatrixByFreePairOfIntervals
  
  MatrixByFreePairOfIntervals( gfs, k, kp )  function
  
  Returns  the  matrix corresponding to the pair of free intervals k and kp in
  the generalized Farey sequence gfs.
  
    Example  
    
    gap> fs_free:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);;
    gap> gfs_free:=GeneralizedFareySequence(fs_free);;
    gap> MatrixByFreePairOfIntervals(gfs_free,2,3);                                                        
    [ [ 3, -2 ], [ 2, -1 ] ]
    
  
  
  4.2-4 GeneratorsByFareySymbol
  
  GeneratorsByFareySymbol( fs )  function
  
  Returns a set of matrices constructed as above.
  
    Example  
    
    gap> fs_eo:=FareySymbolByData([infinity,0,infinity],["even","odd"]);;
    gap> GeneratorsByFareySymbol(last);                                  
    [ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
    gap> GeneratorsByFareySymbol(fs); 
    [ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ]
    gap> GeneratorsByFareySymbol(fs_oo);
    [ [ [ -1, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
    gap> GeneratorsByFareySymbol(fs_free);                                                        
    [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
    
  
  
  4.2-5 GeneratorsOfGroup
  
  GeneratorsOfGroup( G )  function
  
  Returns a set of generators for the finite index group G in SL_2(Z).
  
    Example  
    
    gap> G:=PrincipalCongruenceSubgroup(2);
    <principal congruence subgroup of level 2 in SL_2(Z)>
    gap> FareySymbol(G);
    [ infinity, 0, 1, 2, infinity ]
    [ 2, 1, 1, 2 ]
    gap> GeneratorsOfGroup(G);
    #I  Using the Congruence package for GeneratorsOfGroup ...
    [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
    gap> H:=CongruenceSubgroupGamma0(5);        
    <congruence subgroup CongruenceSubgroupGamma_0(5) in SL_2(Z)>
    gap> GeneratorsOfGroup(H);
    #I  Using the Congruence package for GeneratorsOfGroup ...
    [ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ]
    gap> I:=IntersectionOfCongruenceSubgroups(PrincipalCongruenceSubgroup(2),CongruenceSubgroupGamma0(3));
    <intersection of congruence subgroups of resulting level 6 in SL_2(Z)>
    gap> FareySymbol(I);
    [ infinity, 0, 1/3, 1/2, 2/3, 1, 4/3, 3/2, 5/3, 2, infinity ]
    [ 1, 5, 4, 3, 2, 2, 3, 4, 5, 1 ]
    gap> GeneratorsOfGroup(I);                                                          
    #I  Using the Congruence package for GeneratorsOfGroup ...
    [ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 11, -2 ], [ 6, -1 ] ], 
      [ [ 19, -8 ], [ 12, -5 ] ], [ [ 17, -10 ], [ 12, -7 ] ], 
      [ [ 7, -6 ], [ 6, -5 ] ] ]
    
  
  
  
  4.3 Other properties derived from Farey symbols
  
  4.3-1 IndexInPSL2ZByFareySymbol
  
  IndexInPSL2ZByFareySymbol( fs )  function
  
  By  Proposition  7.2  in  [Kulkarni],  for  the Farey symbol with underlying
  generalized  Farey sequence [infinity, x0, x1, ..., xn, infinity], the index
  in  PSL_2(Z) is given by the formula d = 3*n + e3, where e3 is the number of
  odd intervals.
  
    Example  
    
    gap> IndexInPSL2ZByFareySymbol(fs);
    6
    gap> IndexInPSL2ZByFareySymbol(fs_oo);
    2
    gap> IndexInPSL2ZByFareySymbol(fs_free);
    6