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].

`‣ 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 ]

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]

`‣ 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 ] ]

`‣ 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 ] ]

`‣ 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 ] ]

`‣ 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 ] ] ]

`‣ 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 ] ] ]

`‣ 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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