Subgroups of $SL_2(\mathbb{Z})$

The repository contains the GAP code and the database used in the paper

Nicolás Mayorga Uruburu, Ariel Pacetti, and Leandro Vendramin
On computing finite index subgroups of $PSL_2(Z)$
https://arxiv.org/abs/2307.01826

Our script uses the GAP packages Yags, congruence, and ToolsForHomalg.

Authors

Ariel Pacetti
Nicolás Mayorga Uruburu
Leandro Vendramin

Constructions

The file kulkarni.g contains functions to construct Kulkarni diagrams up to equivalence by conjugation in $SL_2(\mathbb{Z})$ and $GL_2(\mathbb{Z})$. To do this, first we need to read our main file:

gap> Read("kulkarni.g");

Here we show an example of the calculations related to conjugacy classes in $GL_2(\mathbb{Z})$:

gap> ComputeGL2Classes(10);;
Index: 2, I constructed 1 GL2 representatives of subgroups in 0ms (= 0:00:00.000)
Index: 3, I constructed 2 GL2 representatives of subgroups in 8ms (= 0:00:00.008)
Index: 4, I constructed 2 GL2 representatives of subgroups in 9ms (= 0:00:00.009)
Index: 5, I constructed 1 GL2 representatives of subgroups in 4ms (= 0:00:00.004)
Index: 6, I constructed 8 GL2 representatives of subgroups in 28ms (= 0:00:00.028)
Index: 7, I constructed 4 GL2 representatives of subgroups in 26ms (= 0:00:00.026)
Index: 8, I constructed 6 GL2 representatives of subgroups in 31ms (= 0:00:00.031)
Index: 9, I constructed 12 GL2 representatives of subgroups in 134ms (= 0:00:00.134)
Index: 10, I constructed 19 GL2 representatives of subgroups in 324ms (= 0:00:00.324)
--
I constructed 10 tables in 568ms (= 0:00:00.568)

Here we show an example of the calculations related to conjugacy classes in $SL_2(\mathbb{Z})$:

gap> ComputeSL2Classes(10);;
Index: 2, I constructed 1 SL2 representatives of subgroups in 0ms (= 0:00:00.000)
Index: 3, I constructed 2 SL2 representatives of subgroups in 8ms (= 0:00:00.008)
Index: 4, I constructed 2 SL2 representatives of subgroups in 10ms (= 0:00:00.010)
Index: 5, I constructed 1 SL2 representatives of subgroups in 4ms (= 0:00:00.004)
Index: 6, I constructed 8 SL2 representatives of subgroups in 24ms (= 0:00:00.024)
Index: 7, I constructed 6 SL2 representatives of subgroups in 24ms (= 0:00:00.024)
Index: 8, I constructed 7 SL2 representatives of subgroups in 28ms (= 0:00:00.028)
Index: 9, I constructed 14 SL2 representatives of subgroups in 123ms (= 0:00:00.123)
Index: 10, I constructed 27 SL2 representatives of subgroups in 298ms (= 0:00:00.298)
--
I constructed 10 tables in 524ms (= 0:00:00.524)

The database

We precomputed Kulkarni diagrams of index up to 20. The folder data/ contains GAP files with this database.

Example 1: passports

We will compute the list of passports corresponding to Kulkarni diagrams (up to conjugation in $SL_2(\mathbb{Z})$ corresponding to subgroups of index 8. The corresponding data is located in the GAP file data/uptoSL2_8.g

This file contains contains a list names diagrams of Kulkarni diagrams. To use this file we need to load the GAP package congruence:

gap> LoadPackage("congruence");

This is needed to understand the generalized Farey sequence. Then one reads the corresponding file, in our case uptoSL2_8.g. We also need our script kulkarni.g:

gap> Read("data/uptoSL2_8.g");
gap> Read("kulkarni.g");

Now to obtain the list of passports we just need to run over the elements of the list diagrams and use the function KulkarniDiagram2Passport.

gap> Read("data/uptoSL2_8.g");
gap> for x in diagrams do
> Display(KulkarniDiagram2Passport(x));
> od;
[ (1,2)(3,7)(4,5), (1,3,4)(6,7,8), (1,2,3,8,6,7,4,5) ]
[ (1,2)(3,7)(5,6), (1,3,4)(5,7,8), (1,2,3,8,5,6,7,4) ]
[ (2,7)(3,4)(5,6), (1,2,3)(5,7,8), (1,2,8,5,6,7,3,4) ]
[ (2,6)(3,4)(7,8), (1,2,3)(5,6,7), (1,2,7,8,5,6,3,4) ]
[ (1,2)(3,7)(4,5)(6,8), (1,3,4)(6,8,7), (1,2,3,6,7,4,5) ]
[ (1,2)(3,7)(4,8)(5,6), (1,3,8)(4,5,7), (1,2,3,4)(5,6,7,8) ]
[ (1,8)(2,7)(3,4)(5,6), (1,5,7)(2,3,8), (1,2)(3,4,8,5,6,7) ]

Example 2a: generalized Farey symbols

We will show the list of generalized Farey symbols corresponding to subgroups (up to conjugation in $SL_2(\mathbb{Z})$ of index 8. As before, first we need to do some technical stuff:

gap> LoadPackage("congruence");
gap> Read("kulkarni.g");

We also need to load the part of the database we want to use:

gap> Read("data/uptoSL2_8.g");

Our Kulkarni diagrams contain the generalized Farey sequence:

gap> for x in diagrams do
> Display(x!.farey_symbol);
> od;
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "odd", "odd", "even", "even" ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "odd", "even", "odd", "even" ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "even", "odd", "odd", "even" ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "even", "odd", "even", "odd" ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "odd", "odd", 5, 5 ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ "odd", 4, "odd", 4 ] ) 
FareySymbolByData( [ infinity, 0, 1, 2, infinity ], [ 3, "odd", "odd", 3 ] )

Example 2b: generalized Farey symbols

gap> Read("data/uptoSL2_12.g");
gap> for x in diagrams do
> Display(x!.farey_symbol);
> od;
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "even", "even", "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", "even", "odd", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", "even", "even", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "even", "odd", "even", "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "even", "odd", "odd", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "even", "odd", "even", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "even", "even", "odd", "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", "odd", "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "even", "odd", "odd", "odd", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "even", "odd", "odd", "even", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", 7, 7, "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", 6, "odd", 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", 5, 5, "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ 4, "odd", 4, "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ 4, 4, "odd", "odd", "odd" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, infinity ], [ "odd", "odd", "odd", 5, 5 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", "even", "even", "even",  "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", "even", "even", "even",  "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ "even", "even", "even", "even", "even",  "even" ] )
FareySymbolByData( [ infinity, 0, 1/3, 1/2, 1, 2, infinity ], [ "even", "even", "even", "even", "even",  "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", 9, 9, "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", "even", 8, "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", "even", 7, 7, "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, "even", "even", 5, "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", 7, "even", 7, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", "even", "even", 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", 8, "even", "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, "even", "even", "even", 7 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", "even", 6, "even", 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, "even", "even", 7, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", "even", 6, 6, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, "even", "even", 5, "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ "even", "even", "even", 9, 9, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ "even", "even", 8, 8, "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ "even", 7, 7, "even", "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, "even", 5, "even", "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, 5, "even", "even", "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, "even", "even", "even", "even", 5 ] )
FareySymbolByData( [ infinity, 0, 1/3, 1/2, 1, 2, infinity ], [ "even", 9, 9, "even", "even", "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", 9, 9, 7, 7 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", 8, 7, 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", "even", 7, 8, 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", 6, 9, 9, "even", 6 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", 6, 8, 6, "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", 6, 6, 8, "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, "even", 9, 9, "even", 5 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, "even", 8, 5, "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, "even", 5, 8, "even", 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", 6, 9, 9, 6, "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ "even", 6, 6, 7, 7, "even" ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, "even", 9, 9, 5, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, 7, "even", 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, 8, "even", 8, 7 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, 8, "even", 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", 8, 6, 8, 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", "even", 8, 6, 6, 8 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, "even", 5, "even", 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, "even", 6, 7, 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ "even", 7, "even", 6, 6, 7 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 5, "even", "even", 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, "even", "even", 5, 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, "even", 5, 6, "even", 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 5, "even", 6, "even", 6 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 5, "even", 6, 6, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ "even", 7, 7, 9, 9, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, "even", 5, 9, 9, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, 5, "even", 9, 9, "even" ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, "even", "even", 9, 9, 5 ] )
FareySymbolByData( [ infinity, 0, 1/3, 1/2, 1, 2, infinity ], [ "even", 9, 9, "even", 7, 7 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 5, 9, 9, 7, 7 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 5, 8, 7, 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 5, 7, 8, 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 6, 7, 5, 7, 6 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 6, 8, 5, 6, 8 ] )
FareySymbolByData( [ infinity, 0, 1, 3/2, 2, 3, infinity ], [ 5, 6, 6, 7, 7, 5 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 7, 7, 5, 9, 9 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 7, 8, 5, 8, 7 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 7, 8, 5, 7, 8 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 2, 3, infinity ], [ 5, 5, 8, 6, 6, 8 ] )
FareySymbolByData( [ infinity, 0, 1/2, 1, 3/2, 2, infinity ], [ 5, 7, 7, 9, 9, 5 ] )

Name		Name	Last commit message	Last commit date
Latest commit History 39 Commits
data		data
graph		graph
log		log
641409582.svg		641409582.svg
README.md		README.md
countGL2.g		countGL2.g
countGL2.log		countGL2.log
countSL2.g		countSL2.g
countSL2.log		countSL2.log
create.g		create.g
graph.g		graph.g
histogram.g		histogram.g
histogram.log		histogram.log
kulkarni.g		kulkarni.g
table.g		table.g
table.log		table.log

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Subgroups of $SL_2(\mathbb{Z})$

Authors

Constructions

The database

Example 1: passports

Example 2a: generalized Farey symbols

Example 2b: generalized Farey symbols

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Subgroups of $SL_2(\mathbb{Z})$

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Constructions

The database

Example 1: passports

Example 2a: generalized Farey symbols

Example 2b: generalized Farey symbols

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