Parametrizing Fuchsian Subgroups of the Bianchi Groups

Maclachlan, C. ; Reid, A. W.

Canadian journal of mathematics, Tome 43 (1991) no. 1, pp. 158-181

Résumé

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

Détail

DOI : 10.4153/CJM-1991-009-1

Mots-clés : 11F06, 20H10

Bibliographie

[1] 1. Beardon, A.F., The Geometry of discrete groups, Graduate texts in Math 91, Springer Verlag (1983). Google Scholar

[2] 2. Borel, A., Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Sc. Norm. Sup. Pisa 8(1981), 1–33. Google Scholar

[3] 3. Borel, A. and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann of Math. 75(1962), 485–535. Google Scholar

[4] 4. Cassete, J.W.S., Rational Quadratic Forms. Academic Press, London (1978). Google Scholar

[5] 5. Chinburg, T. and Friedman, E., Torsion in arithmetic subgroups of forms 0/PGL2. (Preprint). Google Scholar

[6] 6. Elstrodt, J., Grunewald, F. and Mennicke, J., A Survey of some aspects of Siegel's theory, Manuscript, Bielefeld (1985). Google Scholar

[7] 7. Fine, B., The structure of?SL2(R), Ann. of Math. Studies 79(1974), 145–170. Google Scholar

[8] 8. Fine, B., Fuchsian subgroups of the Picard group, Can. J. Math 28(1976), 481–486. Google Scholar

[9] 9. Fine, B., Fuchsian embeddings in the Bianchi groups, Can. J. Math. 39(1987), 1434–1445. Google Scholar

[10] 10. Fine, B. and Frohman, C., Some amalgam structures for Bianchi groups, Proc. A.M.S. 102(1988), 221–229. Google Scholar

[11] 11. Fine, B. and Newman, M., Normal subgroup structure of the Picard group, Trans. A.M.S. 302(1987), 769- 786. Google Scholar

[12] 12. Greenberg, L., On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, Amer.J. Math. 89(1967), 56–68. Google Scholar

[13] 13. Harding, S., Ph.D. thesis. Univ. of Southampton, 1985. Google Scholar

[14] 14. Helling, H., Bestimmung der Kommersurabilitätsklasse der Hilbertschen Modulgruppe, Math. Zeit. 92(1966), 269–280. Google Scholar

[15] 15. Hoare, A.H.M., Karass, A. and Solitar, D., Subgroups of infinite index in Fuchsian groups, Math. Zeit 125(1972), 59–69. Google Scholar

[16] 16. Humbert, P., Reduction de formes quadratiques dans un corps algébrique fini, Comment. Math. Helv. 23(1949), 50–63. Google Scholar

[17] 17. Lozano, M.T., Arcbodies, Math. Proc. Camb. Phil. Soc. 94(1983), 253–260. Google Scholar

[18] 18. Maclachlan, C., Fuchsian subgroups of the groups PSL2(Od). In Low-Dimensional Topology and Kleinian groups, ed. D. B. A. Epstein. L.M.S. Lecture note series 112(1986), 305–311. Google Scholar

[19] 19. Maclachlan, C. and Reid, A.W., Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Camb. Phil. Soc. 102(1987), 251–258. Google Scholar

[20] 20. Maclachlan, C. and Reid, A.W., Maclachlan and Reid, A.W., The arithmetic structure of tetrahedral groups of hyperbolic isometries, Mathematika 36(1989). Google Scholar

[21] 21. Reid, A.W., Ph.D. thesis, Univ. of Aberdeen, 1987. Google Scholar

[22] 22. Maclachlan, and Reid, A.W., Totally geodesic surfaces in hyperbolic 3-manifolds, to appear in Proc. Edinburgh Math. Soc. Google Scholar

[23] 23. Rohlfs, J., On the cuspidal cohomology of the Bianchi modular groups, Math. Zeit. 188(1985), 253–269. Google Scholar

[24] 24. Scott, G.P., Subgroups of surface groups are almost geometric, J. Lond. Math. Soc. (2) 17(1978), 555–565. Google Scholar

[25] 25. Siegel, C.L., Über die analytische theorie der quadratischen formen I, II, III, Ann. of Math. 36(1935),527- 606. Ibid. 37(1936), 230–267. Ibid. 38(1937), 212–291. Google Scholar

[26] 26. Singerman, D., Subgroups of Fuchsian groups and finite permutation groups, Bull. London Math. Soc. 2(1970), 319–323. Google Scholar

[27] 27. Takeuchi, K., On some discrete subgroups of SL2( ), J. Fac. Sci. Univ. Tokyo Sec. 116(1969), 97–100. Google Scholar

[28] 28. Takeuchi, K., Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sec. 124(1977), 201–212. Google Scholar

[29] 29. Thurston, W.P., The geometry and topology of 3-manifolds. Lecture notes, Princeton Univ., 1978. Google Scholar

[30] 30. Vignéras, M.-F., Arithmétique des Algèbres de Quaternions, L.N.M. 800, Springer-Verlag, 1980. Google Scholar

[31] 31. Wielenberg, N.J., The structure of certain subgroups of the Picard group, Math. Proc. Camb. Phil. Soc. 84(1978), 427–436. Google Scholar

[32] 32. Zimmert, R., Zür SL2 der ganzen Zahlen eines imaginar-quadratischen Zahlkorpers, Invent. Math. 19(1973), 73–82. Google Scholar