The finite subgroups of maximal arithmetic kleinian groups
Annales de l'Institut Fourier, Volume 50 (2000) no. 6, pp. 1765-1798.

Given a maximal arithmetic Kleinian group Γ PGL (2,), we compute its finite subgroups in terms of the arithmetic data associated to Γ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.

Nous calculons, en fonction des paramètres arithmétiques décris par Borel, les sous-groupes finis d’un groupe de Klein arithmétique maximal. Ceci est notamment appliquable à l’étude des 3-variétés arithmétiques hyperboliques.

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     title = {The finite subgroups of maximal arithmetic kleinian groups},
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Chinburg, Ted; Friedman, Eduardo. The finite subgroups of maximal arithmetic kleinian groups. Annales de l'Institut Fourier, Volume 50 (2000) no. 6, pp. 1765-1798. doi : 10.5802/aif.1807. http://archive.numdam.org/articles/10.5802/aif.1807/

[1] A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa, 8 (1981), 1-33. Also in Borel's Oeuvres, Berlin, Springer, 1983. | Numdam | MR | Zbl

[2] T. Chinburg and E. Friedman, An embedding theorem for quaternion algebras J. London Math. Soc., (2) 60 (1999), 33-44. | MR | Zbl

[3] T. Chinburg and E. Friedman, The smallest arithmetic hyperbolic 3-orbifold, Invent. Math., 86 (1986), 507-527. | MR | Zbl

[4] T. Chinburg, E. Friedman, K. Jones and A. W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa (to appear). | Numdam

[5] M. Deuring, Algebren, Springer Verlag, Berlin, 1935. | Zbl

[6] W. Feit, Exceptional subgroups of GL2, Appendix to chapter XI of Introduction to Modular Forms by S. Lang, Berlin, Springer Verlag, 1976.

[7] F. W. Gehring and G. J. Martin, 6-torsion and hyperbolic volume, Proc. Amer. Math. Soc., 117 (1993), 727-735. | MR | Zbl

[8] F. W. Gehring, C. Maclachlan, G. J. Martin and A. W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc., 349 (1997), 3611-3643. | Zbl

[9] K. N. Jones and A. W. Reid, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann., 310 (1998), 235-250. | MR | Zbl

[10] J. Martinet, Petits discriminants des corps de nombres. In J. Armitage, editor, Number Theory Days, 1980 (Exeter 1980). London Math. Soc. Lecture Notes Ser. 56, Cambridge: Cambridge Univ. Press, 1982. | MR | Zbl

[11] J.-P. Serre, Trees, Springer Verlag, Berlin, 1980. | MR | Zbl

[12] L. Washington, Introduction to Cyclotomic Fields, Springer Verlag, Berlin, 1982. | MR | Zbl

[13] M.-F. Vignéras, Arithmétique des algèbres de Quaternions, Lecture Notes in Math. 800, Springer Verlag, Berlin, 1980. | MR | Zbl

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