Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

 Access to full text

 Full (PDF)

1. 1. J. Ax, On Schanuel’s conjecture, Ann. Math. (2)93 (1971), p. 252-268 Zbl0232.10026MR277482
2. 2. A. Baker, Transcendental Number Theory, Cambridge Mathematical Library (1990), Cambridge Univ. Press, Cambridge Zbl0715.11032MR1074572
3. 3. A. Borel, Linear Algebraic Groups, GTM 126 (1991), Springer, Berlin Zbl0726.20030MR1102012
4. 4. A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. IHES69 (1989), p. 119-171 Zbl0707.11032MR1019963
5. 5. A. Borel, J. Tits, Groupes réductifs, Publ. Math. IHES27 (1965), p. 55-151 Zbl0145.17402MR207712
6. 6. M. V. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J.72 (1993), p. 217-239 Zbl0849.12011MR1242885
7. 7. N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, Chap. III, 1972, Chap. IV–VI, 1968. Zbl0483.22001
8. 8. T. Chinburg, E. Hamilton, D. D. Long, A. W. Reid, Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds, Duke Math. J.145 (2008), p. 25-44 Zbl1169.53030MR2451288
9. 9. S. DeBacker, Parametrizing conjugacy classes of maximal unramified tori via Bruhat–Tits theory, Michigan Math. J.54 (2006), p. 157-178 Zbl1118.22005MR2214792
10. 10. J. J. Duistermaat, V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math.29 (1975), p. 39-79 Zbl0307.35071MR405514
11. 11. J. J. Duistermaat, J. A. C. Kolk, V. S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math.52 (1979), p. 27-93 Zbl0434.58019MR532745
12. 12. E. V. Erovenko, A. S. Rapinchuk, Bounded generation of S-arithmetic subgroups of isotropic orthogonal groups over number fields, J. Number Theory119 (2006), p. 28-48 Zbl1168.20020MR2228948
13. 13. R. Gangolli, The length spectra of some compact manifolds, J. Differ. Geom.12 (1977), p. 403-424 Zbl0365.53016MR650997
14. 14. P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, (1995), CRC Press, Boca Raton Zbl0856.58001MR1396308
15. 15. P. Gille, Type des tores maximaux des groupes semi-simples, J. Ramanujan Math. Soc.19 (2004), p. 213-230 Zbl1193.20057MR2139505
16. 16. G. Harder, Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen, Jber. Deutsch. Math.-Verein.70 (1967), p. 182-216 Zbl0194.05701MR242838
17. 17. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, (2001), AMS, Providence Zbl0993.53002MR1834454
18. 18. K. Kariyama, On conjugacy classes of maximal tori in classical groups, J. Algebra125 (1989), p. 133-149 Zbl0682.20031MR1012667
19. 19. C. J. Leininger, D. B. McReynolds, W. Neumann, and A. W. Reid, Length and eigenvalue equivalence, Int. Math. Res. Notices, 24 (2007). Zbl1158.53032MR2377017
20. 20. A. Lubotzky, B. Samuels, U. Vishne, Division algebras and noncommensurable isospectral manifolds, Duke Math. J.135 (2006), p. 361-379 Zbl1123.58020MR2267287
21. 21. G. A. Margulis, Discrete Subgroups of Semi-Simple Lie Groups, (1991), Springer, Berlin Zbl0732.22008MR1090825
22. 22. H. P. McKean, The Selberg trace formula as applied to a compact Riemann surface, Commun. Pure Appl. Math.25 (1972), p. 225-246 Zbl0317.30018MR473166
23. 23. D. Morris, Bounded generation of SL(n,A) (after D. Carter, G. Keller, and E. Paige), N. Y. J. Math.13 (2007), p. 383-421 Zbl1137.20045MR2357719
24. 24. A. Yu. Ol’shanskii, Periodic quotient groups of hyperbolic groups, Math. USSR-Sb.72 (1992), p. 519-541 Zbl0820.20044MR1119008
25. 25. R. S. Pierce, Associative Algebras, GTM 88 (1982), Springer, Berlin Zbl0497.16001MR674652
26. 26. V. P. Platonov, A. S. Rapinchuk, Algebraic Groups and Number Theory, (1994), Academic Press, New York Zbl0841.20046MR1278263
27. 27. G. Prasad, Lattices in semi-simple groups over local fields, in: Adv. in Math. Studies in Algebra and Number Theory, (1979), Academic Press, New York MR535769
28. 28. G. Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France110 (1982), p. 197-202 Zbl0492.20029MR667750
29. 29. G. Prasad, M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups, Ann. Math. (2)96 (1972), p. 296-317 Zbl0245.22013MR302822
30. 30. G. Prasad, A. S. Rapinchuk, Computation of the metaplectic kernel, Publ. Math. IHES84 (1996), p. 91-187 Zbl0941.22019MR1441007
31. 31. G. Prasad, A. S. Rapinchuk, Irreducible tori in semisimple groups, Int. Math. Res. Notices23 (2001), p. 1229-1242 Zbl1057.22025MR1866442
32. 32. G. Prasad, A. S. Rapinchuk, Existence of irreducible ℝ-regular elements in Zariski-dense subgroups, Math. Res. Lett.10 (2003), p. 21-32 Zbl1029.22020MR1960120
33. 33. G. Prasad, A. S. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Math. Res. Lett.12 (2005), p. 239-249 Zbl1072.22009MR2150880
34. 34. G. Prasad, A. S. Rapinchuk, On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior, Adv. Math.207 (2006), p. 646-660 Zbl1132.20024MR2271021
35. 35. G. Prasad and A. S. Rapinchuk, Local-global principles for embedding of fields with involution into simple algebras with involution, available at: arXiv:0806.0596 (to appear in Comment. Math. Helv.). Zbl1223.11047MR2653693
36. 36. M. S. Raghunathan, Discrete Subgroups of Lie Groups, (1972), Springer, Berlin Zbl0254.22005MR507234
37. 37. M. S. Raghunathan, Tori in quasi-split groups, J. Ramanujan Math. Soc.19 (2004), p. 281-287 Zbl1080.20042MR2125504
38. 38. A. W. Reid, Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds, Duke Math. J.65 (1992), p. 215-228 Zbl0776.58040MR1150584
39. 39. J.-P. Serre, Galois Cohomology, (1997), Springer, Berlin Zbl0902.12004MR1466966
40. 40. D. J. Saltman, A note on quaternion algebras (preprint). 
41. 41. T. A. Springer, Linear Algebraic Groups, (1998), Birkhäuser, Basel Zbl1202.20048MR1642713
42. 42. T. Sunada, Riemann coverings and isospectral manifolds, Ann. Math. (2)121 (1985), p. 169-186 Zbl0585.58047MR782558
43. 43. J. Tits, Algebraic and abstract simple groups, Ann. Math. (2)80 (1964), p. 313-329 Zbl0131.26501MR164968
44. 44. J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Groups, Proc. Sympos. Pure Math. 9 (1966), Amer. Math. Soc., Providence Zbl0238.20052MR224710
45. 45. M.-F. Vignéras, Variétés Riemanniennes isospectrales et non isométriques, Ann. Math. (2)112 (1980), p. 21-32 Zbl0445.53026
46. 46. E. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Math. USSR Izv.5 (1971), p. 45-55 Zbl0252.20043MR279206
47. 47. V. E. Voskresenskii, Algebraic Groups and Their Birational Invariants, (1998), AMS, Providence Zbl0974.14034MR1634406
48. 48. B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups, Ann. Math. (2)120 (1984), p. 271-315 Zbl0568.14025MR763908

1. Alan W. Reid, Traces, lengths, axes and commensurability