Statements (228)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:group_of_people
gptkb:mathematical_concept modular group discrete group |
| gptkbp:actionIs |
not free
properly discontinuous |
| gptkbp:actsOn |
gptkb:hyperbolic_plane
gptkb:unit_disk upper half-plane isometries |
| gptkbp:alternativeName |
Baumslag–Solitar_group
Fuchsian_group arithmetic_group congruence_subgroup modular_curve modular_form |
| gptkbp:application |
gptkb:Langlands_program
gptkb:string_theory gptkb:Teichmüller_theory gptkb:monstrous_moonshine modular forms proof of Fermat's Last Theorem automorphic forms moduli spaces geometry of locally symmetric spaces partition function in physics examples in group theory with unusual properties |
| gptkbp:arises_in |
gptkb:Langlands_program
arithmetic geometry modular representation theory |
| gptkbp:automorphismGroup |
modular group or congruence subgroup
|
| gptkbp:category |
gptkb:geometry
gptkb:topology complex analysis group theory number theory |
| gptkbp:class |
gptkb:elementary_Fuchsian_group
gptkb:non-elementary_Fuchsian_group |
| gptkbp:compact |
adding cusps
|
| gptkbp:containsElement |
fractional linear transformation
|
| gptkbp:definedIn |
gptkb:software
complex numbers upper half-plane congruence conditions modulo an integer |
| gptkbp:defines |
A Fuchsian group is a discrete subgroup of PSL(2, R).
A subgroup of a Lie group that is commensurable with the integer points of an algebraic group defined over the rationals. |
| gptkbp:discreteness |
discrete subgroup
|
| gptkbp:example |
gptkb:Trinity
gptkb:j-invariant gptkb:principal_congruence_subgroup gptkb:Eisenstein_series gptkb:Hecke_group gptkb:modular_group_PSL(2,_Z) gptkb:Ramanujan_tau_function gptkb:Gamma_0(N) gptkb:Gamma_1(N) gptkb:Hecke_subgroup X(N) X_0(N) X_1(N) theta function Delta function Gamma(N) |
| gptkbp:field |
gptkb:algebra
gptkb:mathematics group theory number theory |
| gptkbp:fundamentalDomain |
region in upper half-plane bounded by |z|=1, -1/2 < Re(z) < 1/2
|
| gptkbp:generalizes |
gptkb:automorphic_form
gptkb:Drinfeld_modular_form gptkb:Hilbert_modular_form gptkb:Maass_form gptkb:Siegel_modular_form modular function |
| gptkbp:generation |
S: z ↦ -1/z
T: z ↦ z+1 |
| gptkbp:has_genus |
depends on level N
|
| gptkbp:hasApplication |
gptkb:hyperbolic_geometry
gptkb:modular_curves gptkb:string_theory gptkb:Teichmüller_theory gptkb:monstrous_moonshine cryptography dynamical systems modular forms automorphic forms elliptic curves modular functions |
| gptkbp:hasInvariant |
gptkb:lion
fundamental domain limit set Fourier coefficient Hecke eigenvalue |
| gptkbp:hasProperty |
growth condition at infinity
holomorphic transformation law under modular group |
| gptkbp:hasSubfield |
gptkb:algebraic_geometry
gptkb:geometry complex analysis group theory number theory |
| gptkbp:hasSubgroup |
gptkb:SL(2,ℝ)
gptkb:principal_congruence_subgroup gptkb:Hecke_group modular group free group infinite cyclic group free abelian group |
| gptkbp:hasType |
gptkb:Hilbert_modular_form
gptkb:Jacobi_form gptkb:Siegel_modular_form gptkb:cusp_form gptkb:Eisenstein_series theta function |
| gptkbp:heldBy |
two-generator one-relator group
subgroup of SL(2, Z) subgroup of modular group |
| gptkbp:importantFor |
modular forms
automorphic forms |
| gptkbp:indexedIn |
infinite
|
| gptkbp:introducedIn |
1962
|
| gptkbp:isQuotientOf |
gptkb:SL(2,ℤ)
infinite cyclic group finite cyclic group |
| gptkbp:level |
gptkb:integral
|
| gptkbp:namedAfter |
gptkb:Gilbert_Baumslag
gptkb:Lazarus_Fuchs gptkb:Donald_Solitar |
| gptkbp:notableExample |
gptkb:GL(n,Z)
gptkb:SL(n,Z) gptkb:BS(m,_n) BS(1, n) |
| gptkbp:notation |
gptkb:PSL(2,ℤ)
gptkb:SL(2,ℤ)/{±I} |
| gptkbp:orderOfS |
2
|
| gptkbp:orderOfST |
3
|
| gptkbp:parameter |
isomorphism classes of elliptic curves with level structure
|
| gptkbp:presentedBy |
⟨S,T | S^2 = (ST)^3 = 1⟩
BS(m, n) = ⟨ a, b | b⁻¹a^m b = a^n ⟩ |
| gptkbp:preserves |
hyperbolic metric
|
| gptkbp:property |
finitely generated
can be a lattice in Lie group can have property (T) often has finite covolume in Lie group can be Hopfian for some parameters can be amenable for some parameters can be metabelian can be non-Hopfian can be non-Hopfian for some parameters can be non-amenable can be non-amenable for some parameters can be non-residually finite can be residually finite for some parameters can be solvable |
| gptkbp:related_conjecture |
gptkb:Taniyama-Shimura_conjecture
|
| gptkbp:relatedGroup |
composition of transformations
|
| gptkbp:relatedTo |
gptkb:automorphic_form
gptkb:Trinity gptkb:algebraic_geometry gptkb:Farey_sequence gptkb:monodromy_group gptkb:triangle_group_(2,3,∞) gptkb:modular_curves gptkb:Lie_group gptkb:Riemann_surfaces gptkb:Bers_area_theorem gptkb:Fuchsian_differential_equation gptkb:Fuchsian_model gptkb:Möbius_transformation gptkb:Schottky_group gptkb:quasifuchsian_group gptkb:Bass–Serre_theory gptkb:solvable_Baumslag–Solitar_group gptkb:elliptic_curve Fourier series modular forms automorphic forms elliptic curves modular group moduli space continued fractions modular function discrete subgroup HNN extension Hopfian group amenable group metabelian group residually finite group group presentation one-relator group group action on tree group automorphism group endomorphism |
| gptkbp:satisfies |
modular invariance
Fourier expansion |
| gptkbp:seeAlso |
gptkb:Margulis_arithmeticity_theorem
gptkb:Borel–Harish-Chandra_theorem |
| gptkbp:structure |
gptkb:algebraic_geometry
gptkb:Riemannian_manifold |
| gptkbp:studiedBy |
gptkb:Yutaka_Taniyama
gptkb:André_Weil gptkb:Erich_Hecke gptkb:Harish-Chandra gptkb:Henri_Poincaré gptkb:Srinivasa_Ramanujan gptkb:Goro_Shimura gptkb:Langlands_program gptkb:Armand_Borel arithmetic geometry |
| gptkbp:studiedIn |
gptkb:Riemann_surfaces
complex analysis geometric group theory combinatorial group theory |
| gptkbp:type |
gptkb:Kleinian_group
|
| gptkbp:used_in |
proof of Fermat's Last Theorem
|
| gptkbp:usedIn |
counterexamples in group theory
study of amenability study of group actions on trees study of group presentations study of metabelian groups study of non-Hopfian groups study of non-residually finite groups study of solvable groups |
| gptkbp:weight |
gptkb:integral
|
| gptkbp:bfsParent |
gptkb:Trinity
gptkb:group_of_people gptkb:quantum_field_theory |
| gptkbp:bfsLayer |
4
|