Goto Chapter: Top 1 2 3 4 5 Bib Ind

### 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


Goto Chapter: Top 1 2 3 4 5 Bib Ind

generated by GAPDoc2HTML