The congruence subgroup problem

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

Volumn 114, Issue 4, 2004, Pages 299-308

  Raghunathan, M S ^a  

^a TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

Author keywords

Algebraic groups; Arithmetic groups; Congruence groups

Indexed keywords

MAPS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; PROBLEM SOLVING; THEOREM PROVING;

ALGEBRAIC GROUPS; ARITHMETIC GROUPS; CONGRUENCE GROUPS;

SET THEORY;

EID: 10044248552     PISSN: 02534142     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02829437     Document Type: Conference Paper

References (94)

1. Bak A, Le probléme des sous-groupes de congruences et le probléme metaplectique pour les groupes classiques de rang > 1, C.R. Acad. Sci. Paris 292 (1981) 307-310
2. n≥2 of division algebras, J. Algebra 78 (1982) 475-547
3. Bass H, Lazard M and Serre J-P, Sous-groupes d'indices finis dans SL(n, ℤ), Bull. Am. Math. Soc. 70 (1964) 385-392
4. 2n(n ≥ 2), Pub. Math. IHES 33 (1967) 59-137; see also 44 (1974) 241-244
5. 2n(n ≥ 2), Pub. Math. IHES 33 (1967) 59-137; see also 44 (1974) 241-244
6. Borel A, Introduction aux groupes arithmétiques (Paris: Hermann) (1969)
7. Borel A and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math. 75 (1962) 485-535
8. Borel A and Tits J, Groupes Réductifs, Publ. Math. IHES 27 (1965) 55-110
9. Borovoi M V, Abstract simplicity of some simple anisotropic algebraic groups over number fields, Sov. Math. Dokl. 32 (1985) 191-193
10. Birtto J, On defining a subgroup of the special linear group by a congruence, J. Ind. Math. Soc. 40 (1976) 235-243
11. Bruhat F and Tits J, Groups réductifs sur un corps local, I: Données radicielles valuéés, Publ. Math. IHES 41 (1972) 5-251; II: Schémas en groupes. Existence d'une donnée radicielles, Publ. Math. IHES 60 (1984) 5-184
12. Bruhat F and Tits J, Groups réductifs sur un corps local, I: Données radicielles valuéés, Publ. Math. IHES 41 (1972) 5-251; II: Schémas en groupes. Existence d'une donnée radicielles, Publ. Math. IHES 60 (1984) 5-184
13. Ĉernusov V, On the projective simplicity of certain groups of rational points over algebraic number fields, Math. USSR Izv. 34 (1990) 409-423
14. Chevalley C, Sur certains groupes simples, Tohoku J. Math. 7(2) (1955) 14-62
15. Chevalley C, Deux théorèmes d'arithmétiques, J. Math. Soc. Japan 3 (1951) 36-44
16. Corlette K, Archimedean super-rigidity and hyperbolic geoemetry, Ann. Math. 135 (1992) 165-182
17. Delaroche C and Kirillov A, Sur les relations entrel'espace dual d'un groupe et la structure de ses sous-groupes fermés, Exposé 343, Séminaire Bourbaki (1967-68)
18. Deligne P, Extensions centrales non résiduellement finis de groupes arithmétiques, C.R. Acad. Sci. Paris, Ser A-B 287(4) (1978) A203-208
19. Deodhar V V, On central extensions of rational points of algebraic groups, Am. J. Math. 100 (1978) 303-386
20. Dickson L E, Linear group (Leipzig: Teubner) (1901)
21. Dieudonné J, La géometrie des groupes classiques (Berlin: Springer-Verlag) (1955)
22. Fricke R and Klein F, Vorlesungen über die Theorie der automorphen Funktionen, Band I: Die gruppentheoretischen Grundlagen, Band II: Die funktionentheoretischen Ausführungen und die Andwendungen (German) (Stuttgart: B.G. Teubner Verlagsgesellschaft) (1965)
23. Harder G, Minkowskische Reduktionstheorie über Funktionenkörpern, Invent Math. 7 (1969) 33-54
24. Kazhdan D A, On the connection between the dual space of the group with the structure of the closed subgroups, Fund. Anal. Appl. 1 (1967) 71-74 (Russian)
25. Kazhdan D A, Some application of the Weil representation, J. Analyse Mat. 32 (1977) 235-248
26. Kazhdan A A and Bernstein I N, The one-dimensional cohomology of discrete subgroups (Russian), Funktional. Anal. i Pril. 4 (1970) 1-5
27. Kneser M, Orthogonale Gruppen über algebraischen Zahlkörpern, Crelles J. 196 (1956) 213-220
28. Kneser M, Normalteiler ganzzahliger Spingruppen, Crelles J. 311/312 (1979) 191-214
29. Kostant B, On the existence and irreducibility of certain series of representation, Bull. Am. Math. Soc. 75 (1969) 627-642
30. Li J-S, Non-vanishing theorems for the cohomology of some arithmetic quotients, J. Reine Angew. Math. 428 (1992) 111-217
31. Lubotzky A, Group presentation, p-adic analytic groups and lattices in SL(2, ℂ), Ann. Math. 118 (1983) 115-130
32. Margulis G A, Arithmetic properties of discrete groups, Russian Math. Surveys 29 (1974) 107-165
33. Margulis G A, Arithmeticity of non-uniform lattices in weakly non-compact groups (Russian), Funktional Anal. i Prilozen 9 (1975) 35-44
34. Margulis G A, Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Inv. Math. 76 (1984) 93-120
35. Margulis G A, Finiteness of quotient groups of discrete subgroups, Fund. Anal. Appl. 13 (1979) 178-187
36. Margulis G A, On the multiplicative group of a quaternion algebra over a global field, Soviet Math. Dokl. 21 (1980) 780-784
37. e ser.) 2 (1969) 1-62
38. Mennicke J, Finite factor groups of the unimodular groups, Ann. Math. 81 (1965) 31-37
39. Mennicke J, Zur Theorie der Siegeischen Modulgruppe, Math. Annalen. 159 (1965) 115-129
40. Millson J, Real vector bundles with discrete structure group, Topology 18 (1979) 83-89
41. Millson J, On the first Betti number of constant negatively curved manifolds, Ann. Math. 104 (1976) 235-247
42. Moore C C, Extensions and low dimensional cohomology theory of locally compact groups I and II, Trans. Am. Math. Soc. 113 (1964) 40-86; III, Trans. Am. Math. Soc. 221 (1976) 1-38
43. Moore C C, Extensions and low dimensional cohomology theory of locally compact groups I and II, Trans. Am. Math. Soc. 113 (1964) 40-86; III, Trans. Am. Math. Soc. 221 (1976) 1-38
44. Moore C C, Group extensions of p-adic and adelic groups, Publ. Math. IHES 35 (1969) 5-70
45. Platonov V P, The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups, Math. USSR Izv. 3 (1969) 1139-1147
46. Platonov V P, Arithmetic and structure problems in linear algebraic groups, Proc. ICM Vancouver 1 (1974) 471-476
47. Platonov V P, The Tannaka-Artin problem, Sov. Math. Dokl. 16 (1975) 468-473
48. Platonov V P and Rapinchuk A S, Algebraic groups and number theory (Academic Press) (1991)
49. Platonov V P and Rapinchuk A S, On the group of rational points of three dimensional groups, Sov. Math. Dokl. 20 (1979) 693-697
50. Platonov V P and Rapinchuk A S, The multiplicative structure of division algebras over number fields and the Hasse norm principle, Sov. Math. Dokl. 26 (1982) 388-390
51. Prasad G, Strong approximation, Ann. Math. 105 (1977) 553-572
52. Prasad G, A variant of a theorem of Calvin Moore, C.R. Acad. Sci. (Paris) Ser 1302 (1982) 405-408
53. Prasad G and Raghunathan M S, Topological central extensions of semi-simple groups over local fields, Ann. Math. 119 (1984) 143-268
54. Prasad G and Raghunathan M S, On the congruence subgroup problem: Determination of the 'metaplectic kernel', Inv. Math. 71 (1983) 21-42
55. Prasad G and Raghunathan M S, On the Kneser-Tits problem, Comm. Math. Helv. 60 (1985) 107-121
56. Prasad G and Rapinchuk A S, Computation of the metaplectic kernel, Publ. Math. IHES 84 (1997) 91-187
57. Raghunathan M S, On the congruence subgroup problem I, Publ. Math. IHES 46 (1976) 107-161
58. Raghunathan M S, On the congruence subgroup problem II, Inv. Math. 85 (1986) 73-117
59. Raghunathan M S, Torsion in co-compact lattices of spin (2, N), Math. Ann. 266 (1984) 403-419
60. Raghunathan M S, On the group of norm 1 elements in a division algebra, Math. Ann. 279 (1988) 457-484
61. Raghunathan M S, A note on generators for arithmetic subgroups of algebraic groups, Pac. J. Math. 152 (1992) 365-373
62. Raghunathan M S, The congruence subgroup problem, Proceedings of the Hyderabad Conference on Algebraic Groups, Hyderabad, India, Dec. 1989, Manoj Prakashan, India
63. Rapinchuk A S, On the congruence subgroup problem for algebraic groups, Dokl. Akad. Nauk. SSSR 306 (1989) 1304-1307
64. Rapinchuk A S, Multiplicative arithmetic of division algebras over number fields and the metaplectic problem, Math. USSR Izv. 31 (1988) 349-379
65. Rapinchuk A S, Combinatorial theory of arithmetic groups, Preprint (Acad. of Sciences BSSR) (1990)
66. Rapinchuk A S, Congruence subgroup problem for algebraic groups, old and new, Journeés Arithmétiques (Geneva) (1991), Astérisque 209(1) (1992) 73-84
67. Rapinchuk A S, Congruence subgroup problem for algebraic groups, old and new, Journeés Arithmétiques (Geneva) (1991), Astérisque 209(1) (1992) 73-84
68. Rapinchuk A S, Segev Y and Seitz G M, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, J. Am. Math. Soc. 15(4) (2002) 929-978 (electronic)
69. Rapinchuk A S and Segev Y, Valuation-like maps and the congruence subgroup property, Invent. Math. 144(3) (2001) 571-607
70. Segal D, Congruence topologies in commutative rings, Bull. London Math. Soc. 11 (1979) 186-190
71. Segev Y, On finite homomorphic images of the multiplicative group of a division algebra. Ann. Math. (2) 149(1) (1999) 219-251
72. 2, Ann. Math. 92 (1970) 489-527
73. Serre J-P, Sur les groupes de congruence des variet́eś abelienne, Izv. Akad. Nauk. SSSR 28 (1964) 3-18; II, Akad. Nauk. SSSR 35 (1971) 731-735
74. Serre J-P, Sur les groupes de congruence des variet́eś abelienne, Izv. Akad. Nauk. SSSR 28 (1964) 3-18; II, Akad. Nauk. SSSR 35 (1971) 731-735
75. 2, Ann. Math. 92 (1970) 489-527
76. Serre J-P, Sur les groupes de congruence des variet́eś abelienne, Izv. Akad Nauk. SSSR 28 (1964) 3-18; II, Akad. Nauk. SSSR 35 (1971) 731-735
77. Serre J-P, Sur les groupes de congruence des variet́eś abelienne, Izv. Akad Nauk. SSSR 28 (1964) 3-18; II, Akad. Nauk. SSSR 35 (1971) 731-735
78. Steinberg R, Geńeérateurs, ref́lations et revet̂ements de groupes algeb́riques, Colloque de Bruxelles, CNRB (1962) 113-127
79. Sury B, Congruence subgroup problem for anisotropic groups over semilocal rings, Proc. Indian Acad. Sci. (Math. Sci.) 101 (1991) 87-110
80. Swan R G, Generators and relations for certain special linear groups, Adv. Math. 6 (1971) 1-77
81. Tits J, Algebraic and abstract simple groups, Ann. Math. 80 (1964) 313-329
82. Tits J, Classification of algebraic semi-simple groups, Proc. Symp. Pure Math. 9 (1966) (AMS)
83. Tits J, Systems geńerateurs de groupes de congruence, C.R. Acad. Sci. Paris Series A 283 (1976) 693-695
84. Toledo D, Projective varieties with non-residually finite fundamental group, Inst. Hautes Études Sci. Publ. Math. 77 (1993) 103-119
85. Tomanov G, On the congruence subgroup problem for some anisotropic algebraic groups over number fields, Grelles J. 402 (1989) 138-152
86. Tomanov G, Projective simplicity of groups of rational points of simply connected algebraic groups defined over number fields, Topics in Algebra Part 2 (Warsaw) (1998) 455-466, (Banach Centre Publications) 26, Part 2, PWN, Warsaw (1990)
87. Tomanov G, On the group of elements of reduced norm l in a division algebra over a global field (Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 55 (1991) 917; translation in Math. USSR-Izv. 39 (1992) 895-904
88. Tomanov G, On the group of elements of reduced norm l in a division algebra over a global field (Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 55 (1991) 917; translation in Math. USSR-Izv. 39 (1992) 895-904
89. Vaserstein L, The structure of classical arithmetic group of rank greater than one, Math. USSR Sb. 20 (1973) 465-492
90. Venkataramana T N, On super-rigidity and arithmeticity of lattices in semisimple groups, Invent Math. 92 (1988) 255-306
91. Venkataramana T N, On systems of generators of arithmetic subgroups of higher rank groups, Pac. J. Math. 166 (1994) 193-212
92. Wall C T C, On the commutator subgroups of certain unitary groups, J. Algebra 27 (1973) 306-310
93. Wall G E, The structure of a unitary factor group, Publ. Math. IHES 1 (1959)
94. Wang S P, The dual space of a semi simple Lie group, Am. J. Math. 91 (1969) 921-937