GAP (Congruence) - Chapter 4: Farey symbols for congruence subgroups

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4 Farey symbols for congruence subgroups
4.1 Computation of the Farey symbol for a finite index subgroup
4.1-1 FareySymbol
4.2 Computation of generators of a finite index subgroup from its Farey symbol
4.2-1 MatrixByEvenInterval
4.2-2 MatrixByOddInterval
4.2-3 MatrixByFreePairOfIntervals
4.2-4 GeneratorsByFareySymbol
4.2-5 GeneratorsOfGroup
4.3 Other properties derived from Farey symbols
4.3-1 IndexInPSL2ZByFareySymbol

### 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.

    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.

    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.

    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.

    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.

    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 \(SL_2(Z)\).

    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.

    gap> IndexInPSL2ZByFareySymbol(fs);
    6
    gap> IndexInPSL2ZByFareySymbol(fs_oo);
    2
    gap> IndexInPSL2ZByFareySymbol(fs_free);
    6

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