Fuchsian Embeddings in the Bianchi Groups

Fine, Benjamin

Canadian journal of mathematics, Tome 39 (1987) no. 6, pp. 1434-1445

Résumé

If d is a positive square free integer we let Od be the ring of integers in and we let Γd = PSL 2(Od ), the group of linear fractional transformations and entries from Od {if d = 1, ad – bc = ±1}. The Γd are called collectively the Bianchi groups and have been studied extensively both as abstract groups and in automorphic function theory {see references}. Of particular interest has been Γ1 – the Picard group. Group theoretically Γ1, is very similar to the classical modular group M = PSL2(Z) both in its total structure [4, 6], and in the structure of its congruence subgroups [8]. Where Γ1 and M differ greatly is in their action on the complex place C. M is Fuchsian and therefore acts discontinuously in the upper half-plane and every subgroup has the same property.

Détail

DOI : 10.4153/CJM-1987-067-1

Bibliographie

[1] 1. Best, L. A., On torsion-free discrete subgroups of PSL(C) with compact orbit space, Can J. Math. 23 (1971), 451–460. Google Scholar

[2] 2. Brunner, A. M., Frame, M., Lee, Y. W. and Wielenberg, N. J., Classifying torsion-free subgroups of the Picard group, Trans, of the A. M. S. 282 (1984), 205–235. Google Scholar

[3] 3. Brunner, A. M., Lee, Y. W. and Wielenberg, N. J., Polyhedral groups and graph amalgamation products, to appear. Google Scholar | DOI

[4] 4. Fine, B., The structure of PSL(R); R the ring of integers in a Euclidean quadratic imaginary number field, Ph.D. thesis, New York University (1973). Google Scholar | DOI

[5] 5. Fine, B., The structure of PSL(R), Ann. of Math. Study. 79 (1974), 145–170. Google Scholar

[6] 6. Fine, B., The HNN and generalized free and generalized free product structure of certain linear groups, Bull. A.M. S. 81 (1975), 413–416. Google Scholar

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

[8] 8. Fine, B., Congruence subgroups of the Picard group, Can. J. Math. 32 (1979), 1474–1481. Google Scholar

[9] 9. Fine, B., Groups whose torsion-free subgroups are free, Bull. Acad. Sin. 12 (1984), 31–36. Google Scholar

[10] 10. Floge, D., Zur struktur der PSL uber einegen imaginar quadratischen Zahlringen, Math Z. 183 (1983), 255–279. Google Scholar

[11] 11. Grunewald, F. and Schwermer, J., Free non-abelian quotients of SL2 over orders of imaginary quadratic number fields, J. of Alg. 69 (1981), 162–175. Google Scholar

[12] 12. Karrass, A. and Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Tran. A. M. S. 150 (1970), 227–255. Google Scholar

[13] 13. Karrass, A. and Solitar, D., Subgroups of HNN groups and groups with one defining relation, Can. J. Math. 23 (1971), 627–643. Google Scholar

[14] 14. Lehner, J., Discontinuous groups and automorphic functions, Math Surveys 8 (Amer. Math. Soc., Providence R. I., 1964). Google Scholar | DOI

[15] 15. Lyndon, R. and Schupp, P., Combinatorial group theory, (Springer-Verlag, New York, 1977). Google Scholar

[16] 16. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, (Wiley Interscience, New York, 1966). Google Scholar

[17] 17. Maskit, B., On a class of Kleinian groups, Ann. Academia Scient. Fennicae I, Mathematica, Helsinki (1969), 1–7. Google Scholar

[18] 18. Maclachlan, C., Fuchsian subgroups of the groups PSL(Od), to appear. Google Scholar

[19] 19. Mennicke, J., A note on regular coverings of closed orientable surfaces, Proc. Glasgow Math. Soc. 5(1969), 49–66. Google Scholar

[20] 20. Newman, M., Integral matrices, (Academic Press, New York, 1972). Google Scholar

[21] 21. Serre, J. P., Trees, (Springer-Verlag, New York, 1980). Google Scholar | DOI

[22] 22. Swan, R. G., Generators and relations for certain special linear groups, Adv. in Math. 6 (1971), 1–77. Google Scholar

[23] 23. Waldinger, H., On the subgroups of the Picard group, Proc. A. M. S. 16 (1965), 1375–1378. Google Scholar