×

Discreteness criteria of Möbius groups of high dimensions and convergence theorems of Kleinian groups. (English) Zbl 0980.20040

Generalizing the result of T. Jørgensen the authors give discreteness criteria for Möbius groups of high dimensions, especially criteria which allow to get discreteness from the discreteness of the two-generator subgroups.

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20H20 Other matrix groups over fields
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20F05 Generators, relations, and presentations of groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahlfors, L. V., Möbius transformations and Clifford numbers, Differential Geometry and Complex Analysis (1985), p. 65-73 · Zbl 0569.30040
[2] Beardon, A. F., The Geometry of Discrete Groups (1983), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0633.30044
[3] Beardon, A. F., Some remarks on non-discrete Möbius groups, Ann. Acad. Sci. Fenn. Math., 21, 69-79 (1996) · Zbl 0856.30032
[4] Chuckrow, V., On Schotty groups with applications to Kleinian groups, Ann. of Math., 88, 47-61 (1968) · Zbl 0186.40603
[5] Gehring, F. W.; Martin, G. J., Discrete quasiconformal groups, Proc. London Math. Soc. (3), 55, 331-358 (1987) · Zbl 0628.30027
[6] Jørgensen, T., On discrete groups of Möbius transformations, Amer. J. Math., 98, 739-749 (1976) · Zbl 0336.30007
[7] Jørgensen, T., A note on subgroups of SL(2, C), Quart. J. Math. Oxford, 28, 209-212 (1977)
[8] Marden, A., Schotty groups and circles, Contributions to Analysis (1974), Academic Press: Academic Press New York, p. 237-278
[9] Martin, G. J., On discrete Möbius groups in all dimensions, Acta Math., 163, 253-289 (1989) · Zbl 0698.20037
[10] Martin, G. J., On the discrete isometry groups of negative curvatures, Pacific J. Math., 160, 109-127 (1993) · Zbl 0822.57026
[11] Sullivan, D., Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math., 155, 243-260 (1985) · Zbl 0606.30044
[12] Tukia, P., Differentiability and rigity of Möbius groups, Invent. Math., 155, 557-578 (1985) · Zbl 0564.30033
[13] Tukia, P., Limits of some Teichmuller-like mappings, Rep. Depart. Math. Univ. Helsinki, 211, 1-50 (1999)
[14] Tukia, P., Convergence groups and Gromov’s metric hyperbolic space, New Zealand J. Math., 23, 157-187 (1994) · Zbl 0855.30036
[15] Tukia, P.; Väisälä, J., A remark on 1-quasiconformal maps, Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 561-562 (1985) · Zbl 0533.30019
[17] Väisälä, J., Lectures on n dimensional quasiconformal mappings, Lecture Notes in Math. (1971), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0221.30031
[18] Wang, X., Classification, generating system and discreteness criterion of Möbius groups in \(\textbf{R}^n \), Acta Math. Sinica, 38, 38-44 (1995)
[19] Wang, X.; Yang, W., Discreteness criterion for subgroups in SL(2, C), Math. Proc. Cambridge Philos. Soc., 124, 51-55 (1998)
[20] Wang, X.; Yang, W., Isometry groups of Hadamard manifold, Complex Variables, 38, 29-34 (1999) · Zbl 1007.30027
[21] Wang, X.; Yang, W., Dense subgroups and discrete subgroups in SL(2, C), Quart. J. Math. Oxford Ser. (2), 50, 517-521 (1999) · Zbl 0941.20055
[22] Waterman, P., Möbius transformations in several dimensions, Adv. Math., 101, 87-113 (1993) · Zbl 0793.15019
[23] Wielenberg, N., Discrete Möbius groups: Fundamental polyhedra and convergence, Amer. J. Math., 99, 861-877 (1977) · Zbl 0373.57024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.