A generalization of the Shimizu-Leutbecher and Jørgensen inequalities to Möbius transformations in $\textbf {R}^ N$
HTML articles powered by AMS MathViewer
- by Sa’ar Hersonsky PDF
- Proc. Amer. Math. Soc. 121 (1994), 209-215 Request permission
Abstract:
We give a generalization of the Shimizu-Leutbecher inequality and a partial generalization of the Jorgensen inequality to Möbius transformations in ${{\mathbf {R}}^N}$ using the Clifford algebra and the Vahlen group.References
- Lars V. Ahlfors, Möbius transformations and Clifford numbers, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 65–73. MR 780036 —, On the fixed points of Möbius transformations in ${{\mathbf {R}}^N}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985).
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- Shmuel Friedland and Sa’ar Hersonsky, Jorgensen’s inequality for discrete groups in normed algebras, Duke Math. J. 69 (1993), no. 3, 593–614. MR 1208812, DOI 10.1215/S0012-7094-93-06924-4 S. Hersonsky, Volume estimates for discrete groups of $\text {Iso}^ + ({H^n})$ having parabolic elements, Michigan Math. J. (to appear).
- Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 427627, DOI 10.2307/2373814
- G. J. Martin, On discrete Möbius groups in all dimensions: a generalization of Jørgensen’s inequality, Acta Math. 163 (1989), no. 3-4, 253–289. MR 1032075, DOI 10.1007/BF02392737
- Hideo Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33–71. MR 145106, DOI 10.2307/1970201
- Armin Leutbecher, Über Spitzen diskontinuierlicher Gruppen von lineargebrochenen Transformationen, Math. Z. 100 (1967), 183–200 (German). MR 214763, DOI 10.1007/BF01109804
- K. Th. Vahlen, Ueber Bewegungen und complexe Zahlen, Math. Ann. 55 (1902), no. 4, 585–593 (German). MR 1511164, DOI 10.1007/BF01450354
- Masaaki Wada, Conjugacy invariants of Möbius transformations, Complex Variables Theory Appl. 15 (1990), no. 2, 125–133. MR 1058518, DOI 10.1080/17476939008814442
- P. L. Waterman, Möbius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87–113. MR 1239454, DOI 10.1006/aima.1993.1043
- Norbert J. Wielenberg, Discrete Moebius groups: fundamental polyhedra and convergence, Amer. J. Math. 99 (1977), no. 4, 861–877. MR 477035, DOI 10.2307/2373869
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 209-215
- MSC: Primary 30F40
- DOI: https://doi.org/10.1090/S0002-9939-1994-1182701-0
- MathSciNet review: 1182701