For representing finite-index subgroups of SL_2(ℤ), this package introduces the new object ModularSubgroup. As stated in the introduction, a ModularSubgroup essentially consists of the two permutations σ_S and σ_T describing the coset graph with respect to the generators S and T (with the convention that 1 corresponds to the identity coset). So explicitly specifying these permutations is the canonical way to construct a ModularSubgroup.
Though you might not always have a coset graph of your subgroup at hand, but rather a list of generating matrices. Therefore we implement multiple constructors for ModularSubgroup: three that take as input two permutations describing the coset graph with respect to different pairs of generators of SL_2(ℤ), and one that takes a list of SL_2(ℤ) matrices as generators.
‣ ModularSubgroupViaRightAction( s, t ) | ( operation ) |
Returns: A modular subgroup.
Constructs a ModularSubgroup object corresponding to the finite-index subgroup of SL_2(ℤ) described by the permutations s and t.
This constructor tests if the given permutations actually describe the coset action of the matrices
[ 0 -1 ] [ 1 1 ]
S = [ 1 0 ], T = [ 0 1 ]
by checking that they act transitively and satisfy the relations
s^4 = (s^3 t)^3 = s^2 t s^{-2} t^{-1} = 1
Upon creation, the cosets are renamed in a standardized way to make the internal interaction with existing GAP methods easier. (The fact that 1 corresponds to the identity coset is not changed by this)
gap> G := ModularSubgroupViaRightAction( > (1,2)(3,4)(5,6)(7,8)(9,10), > (1,4)(2,5,9,10,8)(3,7,6)); <modular subgroup of index 10>
‣ ModularSubgroupViaLeftAction( s, t ) | ( operation ) |
Returns: A modular subgroup.
Analogous to ModularSubgroupViaRightAction, but with left coset actions.
‣ ModularSubgroupSTViaRightAction( s, t ) | ( operation ) |
Returns: A modular subgroup.
Synonymous for ModularSubgroupViaRightAction (2.1-1) (see above).
‣ ModularSubgroupSTViaLeftAction( s, t ) | ( operation ) |
Returns: A modular subgroup.
Synonymous for ModularSubgroupViaLeftAction (2.1-1) (see above).
‣ ModularSubgroupRTViaRightAction( r, t ) | ( operation ) |
Returns: A modular subgroup.
Constructs a ModularSubgroup object corresponding to the finite-index subgroup of SL_2(ℤ) determined by the permutations r and t which describe the action of the matrices
[ 1 0 ] [ 1 1 ]
R = [ 1 1 ] T = [ 0 1 ]
on the right cosets.
A check is performed if the permutations actually describe such an action on the cosets of some subgroup.
Upon creation, the cosets are renamed in a standardized way to make the internal interaction with existing GAP methods easier. (The fact that 1 corresponds to the identity coset is not changed by this)
gap> G := ModularSubgroupRTViaRightAction( > (1,9,8,10,7)(2,6)(3,4,5), > (1,3)(2,4,8,10,5)(6,9,7)); <modular subgroup of index 10>
‣ ModularSubgroupRTViaLeftAction( r, t ) | ( operation ) |
Returns: A modular subgroup.
Analogous to ModularSubgroupRTViaRightAction, but with left coset actions.
‣ ModularSubgroupSJViaRightAction( s, j ) | ( operation ) |
Returns: A modular subgroup.
Constructs a ModularSubgroup object corresponding to the finite-index subgroup of SL_2(ℤ) determined by the permutations s and j which describe the action of the matrices
[ 0 -1 ] [ 0 1 ]
S = [ 1 0 ] J = [ -1 1 ]
on the right cosets.
A check is performed if the permutations actually describe such an action on the cosets of some subgroup.
Upon creation, the cosets are renamed in a standardized way to make the internal interaction with existing GAP methods easier. (The fact that 1 corresponds to the identity coset is not changed by this)
gap> G := ModularSubgroupSJViaRightAction( > (1,2)(3,6)(4,7)(5,9)(8,10), > (1,5,6)(2,3,7)(4,9,10)); <modular subgroup of index 10>
‣ ModularSubgrouSJViaLeftAction( s, j ) | ( operation ) |
Returns: A modular subgroup.
Analogous to ModularSubgroupSJViaRightAction, but with left coset actions.
‣ ModularSubgroup( gens ) | ( operation ) |
Returns: A modular subgroup.
Constructs a ModularSubgroup object corresponding to the finite-index subgroup of SL_2(ℤ) generated by the matrices in gens.
No test is performed to check if the generated subgroup actually has finite index!
This constructor implicitly computes a coset table of the subgroup. Hence it might be slow for very large index subgroups.
gap> G := ModularSubgroup([ > [[1,2], [0,1]], > [[1,0], [2,1]], > [[-1,0], [0,-1]] > ]); <modular subgroup of index 6>
‣ SRightAction( G ) | ( operation ) |
Returns: A permutation.
Returns the permutation σ_S describing the action of the matrix S on the right cosets of G.
‣ SLeftAction( G ) | ( operation ) |
Returns: A permutation.
Analogous to SRightAction, but with left coset actions.
‣ TRightAction( G ) | ( operation ) |
Returns: A permutation.
Returns the permutation σ_T describing the action of the matrix T on the right cosets of G.
‣ TLeftAction( G ) | ( operation ) |
Returns: A permutation.
Analogous to TRightAction, but with left coset actions.
‣ RRightAction( G ) | ( operation ) |
Returns: A permutation.
Returns the permutation σ_R describing the action of the matrix R on the right cosets of G.
‣ RLeftAction( G ) | ( operation ) |
Returns: A permutation.
Analogous to RRightAction, but with left coset actions.
‣ JRightAction( G ) | ( operation ) |
Returns: A permutation.
Returns the permutation σ_J describing the action of the matrix J on the right cosets of G.
‣ JLeftAction( G ) | ( operation ) |
Returns: A permutation.
Analogous to JRightAction, but with left coset actions.
‣ CosetRightActionOf( A, G ) | ( operation ) |
Returns: A permutation.
Returns the permutation σ_A describing the action of the matrix A ∈ SL_2(ℤ) on the right cosets of G.
‣ CosetLeftActionOf( A, G ) | ( operation ) |
Returns: A permutation.
Analogous to CosetRightActionOf, but with left coset actions.
‣ Index( G ) | ( attribute ) |
Returns: A natural number.
For a given modular subgroup G this method returns its index in SL_2(ℤ). As G is internally stored as permutations (s,t) this is just LargestMovedPoint([s,t]) (or 1 if the permutations are trivial).
gap> G := ModularSubgroup((1,2)(3,5)(4,6), (1,3)(2,4)(5,6)); <modular subgroup of index 6> gap> Index(G); 6
‣ GeneralizedLevel( G ) | ( attribute ) |
Returns: A natural number.
This method calculates the general Wohlfahrt level (i.e. the lowest common multiple of all cusp widths) of G as defined in [Woh64].
gap> G := ModularSubgroup((1,2)(3,5)(4,6), (1,3)(2,4)(5,6)); <modular subgroup of index 6> gap> GeneralizedLevel(G); 2
‣ RightCosetRepresentatives( G ) | ( attribute ) |
Returns: A list of words.
This function returns a list of representatives of the (right) cosets of G as words in S and T.
gap> G := ModularSubgroup((1,2),(2,3)); <modular subgroup of index 3> gap> RightCosetRepresentatives(G); [ <identity ...>, S, S*T ]
‣ LeftCosetRepresentatives( G ) | ( attribute ) |
Returns: A list of words.
This function returns a list of representatives of the (left) cosets of G as words in S and T.
‣ WordGeneratorsOfGroup( G ) | ( attribute ) |
Returns: A list of words.
Calculates a list of generators (as words in S and T) of G. This list might include redundant generators (or even duplicates).
gap> G := ModularSubgroup((1,2)(3,5)(4,6), (1,3)(2,4)(5,6)); <modular subgroup of index 6> gap> WordGeneratorsOfGroup(G); [ S^-2, T^-2, S*T^-2*S^-1 ]
‣ GeneratorsOfGroup( G ) | ( attribute ) |
Returns: A list of matrices.
Calculates a list of generator matrices of G. This list might include redundant generators (or even duplicates).
gap> G := ModularSubgroup((1,2)(3,5)(4,6), (1,3)(2,4)(5,6)); <modular subgroup of index 6> gap> GeneratorsOfGroup(G); [ [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, -2 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 2, 1 ] ] ]
‣ IsCongruence( G ) | ( attribute ) |
Returns: True or false.
This method test whether a given modular subgroup G is a congruence subgroup. It is essentially an implementation of an algorithm described in [HL14].
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> IsCongruence(G); true
‣ Cusps( G ) | ( attribute ) |
Returns: A list of rational numbers and infinity.
This method computes a list of inequivalent cusp representatives with respect to G.
gap> G := ModularSubgroup( > (1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,17)(16,19)(18,21)(20,23)(22,24), > (1,3,7,4)(2,5)(6,9,8,12,14,10)(11,13,16,20,18,15)(17,21,22,19)(23,24) > ); <modular subgroup of index 24> gap> Cusps(G); [ infinity, 0, 1, 2, 3/2, 5/3 ]
‣ CuspWidth( c, G ) | ( operation ) |
Returns: A natural number.
This method takes as input a cusp c (a rational number or infinity) and a modular group G and calculates the width of this cusp with respect to G.
gap> G := ModularSubgroup( > (1,2,6,3)(4,11,15,12)(5,13,16,14)(7,17,9,18)(8,19,10,20)(21,24,22,23), > (1,4,5)(2,7,8)(3,9,10)(6,15,16)(11,20,21)(12,19,22)(13,23,17)(14,24,18) > ); <modular subgroup of index 24> gap> CuspWidth(-1, G); 3 gap> CuspWidth(infinity, G); 3
‣ CuspsEquivalent( p, q, G ) | ( operation ) |
Returns: True or false.
Takes two cusps p and q and a modular subgroup G and checks if they are equivalent modulo G, i.e. if there exists a matrix A ∈ G with Ap = q.
gap> G := ModularSubgroup( > (1,2,6,3)(4,11,15,12)(5,13,16,14)(7,17,9,18)(8,19,10,20)(21,24,22,23), > (1,4,5)(2,7,8)(3,9,10)(6,15,16)(11,20,21)(12,19,22)(13,23,17)(14,24,18) > ); <modular subgroup of index 24> gap> CuspsEquivalent(infinity, 1, G); false gap> CuspsEquivalent(-1, 1/2, G); true
‣ CosetRepresentativeOfCusp( c, G ) | ( operation ) |
Returns: A word in S and T.
For a cusp c this function returns a right coset representative A of G such that A ∞ and c are equivalent with respect to G.
gap> G := ModularSubgroup( > (1,2,6,3)(4,11,15,12)(5,13,16,14)(7,17,9,18)(8,19,10,20)(21,24,22,23), > (1,4,5)(2,7,8)(3,9,10)(6,15,16)(11,20,21)(12,19,22)(13,23,17)(14,24,18) > ); <modular subgroup of index 24> gap> CosetRepresentativeOfCusp(4, G); T*S
‣ IndexModN( G, N ) | ( operation ) |
Returns: A natural number.
For a modular subgroup G and a natural number N this method calculates the index of the projection barG of G in SL_2(ℤ/Nℤ).
gap> G := ModularSubgroup((1,2)(3,5)(4,6), (1,3)(2,4)(5,6)); <modular subgroup of index 6> gap> IndexModN(G, 2); 6
‣ Deficiency( G, N ) | ( operation ) |
Returns: A natural number.
For a modular subgroup G and a natural number N this method calculates the so-called deficiency of G from being a congruence subgroup of level N.
The deficiency of a finite-index subgroup Γ of SL_2(ℤ) was introduced in [Wei15]. It is defined as the index [Γ(N) : Γ(N) ∩ Γ] where Γ(N) is the principal congruence subgroup of level N.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> Deficiency(G, 2); 2 gap> Deficiency(G, 4); 1
‣ Deficiency( G ) | ( attribute ) |
Returns: A natural number.
Shorthand for Deficiency(G, GeneralizedLevel(G)).
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> Deficiency(G); 1 gap> Deficiency(G, GeneralizedLevel(G)); 1
‣ Projectivization( G ) | ( operation ) |
Returns: A projective modular subgroup.
For a given modular subgroup G this function calculates its image barG under the projection π : SL_2(ℤ) → PSL_2(ℤ).
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> Projectivization(G); <projective modular subgroup of index 6>
‣ ConjugateGroup( G, A ) | ( operation ) |
Returns: A ModularSubgroup.
Conjugates the group G by the SL_2(ℤ) matrix A and returns the group A^-1*G*A.
‣ NormalCore( G ) | ( attribute ) |
Returns: A modular subgroup.
Calculates the normal core of G in SL_2(ℤ), i.e. the maximal subgroup of G that is normal in SL_2(ℤ).
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> NormalCore(G); <modular subgroup of index 48>
‣ QuotientByNormalCore( G ) | ( attribute ) |
Returns: A finite group.
Calculates the quotient of SL_2(ℤ) by the normal core of G.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> QuotientByNormalCore(G); <permutation group with 2 generators>
‣ AssociatedCharacterTable( G ) | ( attribute ) |
Returns: A character table.
Returns the character table of SL_2(ℤ)/N where N is the normal core of G.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> AssociatedCharacterTable(G); CharacterTable( <permutation group of size 48 with 2 generators> )
‣ IsElementOf( A, G ) | ( operation ) |
Returns: True or false.
This function checks if a given matrix A is an element of the modular subgroup G.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> IsElementOf([[-1,0],[0,-1]], G); false gap> IsElementOf([[1,4],[0,1]], G); true
‣ IsWordElementOf( s, G ) | ( operation ) |
Returns: True or false.
This function checks if a given string s represents an element of the modular subgroup G, written in the generators S and T.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> IsWordElementOf("S^4 * T^2", G); true
‣ IsWordElementOf( w, G ) | ( operation ) |
Returns: True or false.
This function checks if a given word w in the generators S and T represents an element of the modular subgroup G.
gap> G := ModularSubgroup([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); <modular subgroup of index 12> gap> F := FreeGroup("S", "T"); <free group on the generators [ S, T ]> gap> S := F.1; S gap> T := F.2; T gap> SL2Z := F / ParseRelators([S, T], "S^4, (S^3*T)^3, S^2*T*S^-2*T^-1"); <fp group on the generators [ S, T ]> gap> S := GeneratorsOfGroup(SL2Z)[1]; S gap> T := GeneratorsOfGroup(SL2Z)[2]; T gap> IsWordElementOf(S^4 * T^2, G); true
‣ Genus( G ) | ( attribute ) |
Returns: A non-negative integer.
Computes the genus of the quotient G ∖ ℍ via an algorithm described in [Sch04].
gap> G := ModularSubgroup((1,2),(2,3)); <modular subgroup of index 3> gap> Genus(G); 0
The following functions are mostly helper functions used internally and are only documented for sake of completeness.
‣ DefinesCosetActionST( s, t ) | ( operation ) |
Returns: True or false.
Checks if two given permutations s and t describe the action of the generator matrices S and T on the cosets of some subgroup. This is the case if they satisfy the relations
s^4 = (s^3 t)^3 = s^2 t s^{-2} t^{-1} = 1
and act transitively.
gap> s := (1,2)(3,4)(5,6)(7,8)(9,10);; gap> t := (1,4)(2,5,9,10,8)(3,7,6);; gap> DefinesCosetActionST(s,t); true
‣ DefinesCosetActionRT( r, t ) | ( operation ) |
Returns: True or false.
Checks if two given permutations r and t describe the action of the generator matrices R and T on the cosets of some subgroup. This is the case if they satisfy the relations
(r t^{-1} r)^4 = ((r t^{-1} r)^3 t)^3 = (r t^{-1} r)^2 t (r t^{-1} r)^{-2} t^{-1} = 1
and act transitively.
gap> r := (1,9,8,10,7)(2,6)(3,4,5);; gap> t := (1,3)(2,4,8,10,5)(6,9,7);; gap> DefinesCosetActionRT(r,t); true
‣ DefinesCosetActionSJ( s, j ) | ( operation ) |
Returns: True or false.
Checks if two given permutations s and j describe the action of the generator matrices S and J on the cosets of some subgroup. This is the case if they satisfy the relations
s^4 = (s^3 j^{-1} s^{-1})^3 = s^2 j^{-1} s^{-2} j = 1
and act transitively.
gap> s := (1,2)(3,4)(5,6)(7,8)(9,10);; gap> j := (1,5,6)(2,3,7)(4,9,10);; gap> DefinesCosetActionSJ(s,j); true
‣ RightCosetActionFromGenerators( gens ) | ( operation ) |
Returns: A tuple of permutations.
Takes a list of generator matrices and calculates the right coset graph (as two permutations σ_S and σ_T) of the generated subgroup of SL_2(ℤ).
gap> RightCosetActionFromGenerators([ > [[1,2],[0,1]], > [[1,0],[2,1]] > ]); [ (1,2,5,3)(4,8,10,9)(6,11,7,12), (1,4)(2,6)(3,7)(5,10)(8,12,9,11) ]
‣ LeftCosetActionFromGenerators( gens ) | ( operation ) |
Returns: A tuple of permutations.
Analogous to RightCosetActionFromGenerators, but with left coset actions.
‣ STDecomposition( A ) | ( operation ) |
Returns: A word in S and T.
Takes a matrix A ∈ SL_2(ℤ) and decomposes it into a word in the generator matrices S and T.
gap> M := [ [ 4, 3 ], [ -3, -2 ] ];; gap> STDecomposition(M); S^2*T^-1*S^-1*T^2*S^-1*T^-1*S^-1
‣ RTDecomposition( A ) | ( operation ) |
Returns: A word in R and T.
Takes a matrix A ∈ SL_2(ℤ) and decomposes it into a word in the generator matrices R and T.
gap> M := [ [ 4, 3 ], [ -3, -2 ] ];; gap> RTDecomposition(M); (R*T^-1*R)^2*T^-1*R^-1*(T*R^-1*T)^2*R^-1*T^-1*R^-1*T*R^-1
‣ SJDecomposition( A ) | ( operation ) |
Returns: A word in S and J.
Takes a matrix A ∈ SL_2(ℤ) and decomposes it into a word in the generator matrices S and J.
gap> M := [ [ 4, 3 ], [ -3, -2 ] ];; gap> SJDecomposition(M); S^3*J*(S^-1*J^-1)^2*S^-1*J*S^-1
‣ STDecompositionAsList( A ) | ( operation ) |
Returns: A list representing a word in S and T.
Takes a matrix A ∈ SL_2(ℤ) and decomposes it into a word in the generator matrices S and T. The word is represented as a list in the format [[generator, exponent], ... ]
gap> M := [ [ 4, 3 ], [ -3, -2 ] ];; gap> STDecompositionAsList(M); [ [ "S", 2 ], [ "T", -1 ], [ "S", -1 ], [ "T", 2 ], [ "S", -1 ], [ "T", -1 ], [ "S", -1 ], [ "T", 0 ] ]
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