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On Ford isometric spheres in complex hyperbolic space

Published online by Cambridge University Press:  24 October 2008

John R. Parker
John R. Parker
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL
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Abstract

The complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 3 , May 1994 , pp. 501 - 512
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Beals, M., Fefferman, C. and Grossman, R.. Strictly pseudoconvex domains in ℂ2. Bull. A.M.S. (New Series) 8 (1983), 125–322.CrossRefGoogle Scholar
[2]Beardon, A. F.. On the isometric circles of elements in a Fuchsian group. J. London. Math. Soc. (2) 10 (1975), 329–337.CrossRefGoogle Scholar
[3]Beardon, A. F.. The Geometry of Discrete Groups (Springer, 1983).CrossRefGoogle Scholar
[4]Conway, J. H. and Jones, A. J.. Trigonometric diophantine equations. Acta Arithmetica 30 (1976), 229–240.CrossRefGoogle Scholar
[5]Culler, M.. Lifting representations to covering groups. Adv. Math. 59 (1986), 64–70.CrossRefGoogle Scholar
[6]Cygan, J.. Wiener's test for Brownian motion on the Heisenberg group. Colloquium Math. 39 (1978), 367–373.CrossRefGoogle Scholar
[7]Ford, L. R.. Automorphic Functions (McGraw-Hill, 1929).Google Scholar
[8]Goldman, W. M.. Complex Hyperbolic Geometry, to appear.Google Scholar
[9]Goldman, W. M. and Parker, J. R.. Dirichlet polyhedra for dihedral groups in complex hyperbolic space. J. Geometric Analysis 2 (1992), 517–554.CrossRefGoogle Scholar
[10]Hecke, E.. Über der Bestimmung Dirichletscher Reihen durchihre Funktionalgleichung. Math. Ann. 112 (1935), 664–699.CrossRefGoogle Scholar
[11]Kra, I.. On lifting Kleinian groups to SL(2, ℂ). In Differential geometry and complex analysis, H. E. Rauch memorial volume (ed. Chavel, I., & Farkas, H. M.) (Springer, 1985), 181–193.CrossRefGoogle Scholar
[12]Matelski, J. P.. The classification of discrete 2-generator subgroups of PSL(2, ℂ). Israel J. Math. 42 (1982), 309–317.CrossRefGoogle Scholar
[13]Parker, J. R.. Shimizu's lemma for complex hyperbolic space. Int. J. Math. 3 (1992), 291–308.CrossRefGoogle Scholar
[14]Patterson, S. J.. On the cohomology of Fuchsian groups. Glasgow Math. J. 16 (1975), 123–140.CrossRefGoogle Scholar
[15]Petersson, H.. Zur analytischen theorie der Grenzkreisgruppen. III. Math. Ann. 115 (1938), 518–572.CrossRefGoogle Scholar
[16]Shimizu, H.. On discontinuous groups operating on the product of the upper half planes. Ann. of Math. 77 (1963), 33–71.CrossRefGoogle Scholar