|
gptkbp:instanceOf
|
gptkb:group_of_people
modular group
discrete group
|
|
gptkbp:actsOn
|
upper half-plane
|
|
gptkbp:centralTo
|
{I, -I}
|
|
gptkbp:containsElement
|
2x2 integer matrices with determinant 1
|
|
gptkbp:field
|
integers
|
|
gptkbp:fullName
|
gptkb:Special_Linear_Group_of_2x2_integer_matrices_with_determinant_1
|
|
gptkbp:generation
|
S = [[0, -1], [1, 0]]
T = [[1, 1], [0, 1]]
|
|
gptkbp:hasFiniteIndexSubgroups
|
yes
|
|
gptkbp:hasInfiniteIndexSubgroups
|
yes
|
|
gptkbp:hasProperty
|
orthogonal group
residually finite
congruence subgroups
free subgroup of finite index
|
|
gptkbp:hasSubgroup
|
gptkb:principal_congruence_subgroup
gptkb:SL(2,_R)
gptkb:free_group_of_rank_2
congruence subgroups
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SL(2, Z)
|
|
gptkbp:indexedIn
|
infinite in SL(2, Q)
|
|
gptkbp:isDenseIn
|
SL(2, R) (in Zariski topology)
|
|
gptkbp:isDiscreteIn
|
gptkb:SL(2,_R)
|
|
gptkbp:isFinitelyGenerated
|
true
|
|
gptkbp:isFinitelyPresented
|
true
|
|
gptkbp:isLatticeIn
|
gptkb:SL(2,_R)
|
|
gptkbp:isNonAbelian
|
true
|
|
gptkbp:isQuotientOf
|
gptkb:PSL(2,_Z)
|
|
gptkbp:notation
|
gptkb:SL(2,_ℤ)
|
|
gptkbp:operator
|
matrix multiplication
|
|
gptkbp:order
|
infinite
|
|
gptkbp:presentedBy
|
<S, T | S^4 = I, (ST)^3 = I>
|
|
gptkbp:relatedTo
|
gptkb:Farey_sequence
modular forms
modular group
continued fractions
|
|
gptkbp:usedIn
|
gptkb:algebraic_geometry
gptkb:topology
gptkb:hyperbolic_geometry
gptkb:string_theory
gptkb:monodromy_groups
cryptography
dynamical systems
number theory
automorphic forms
theory of modular forms
theory of Riemann surfaces
modular symbols
modular group actions
modular lattices
|
|
gptkbp:bfsParent
|
gptkb:Modular_form
|
|
gptkbp:bfsLayer
|
5
|